nm0929: right shall we make a start them please am i switched on yeah good er 
just a reminder as always er i have my office hours people are beginning to 
come and see me which is good i get the feeling that the assessed exercise is 
focusing minds which is great er notice when these hours are i sh-, i'm i'm 
available then with a caveat from yesterday unfortunately Murphy's law but 
these are times when you can come anyway you don't need to ask me first can i 
come and see you in your office hours just turn up and the same is true of 
other lecturers and class tutors er as well so do use the office hours if you 
need to and you will of course find as you start revising for your examinations 
that you need you need to so get used to it earlier rather than later where are 
we in the things that we have to discuss well we've got three more lect-, three 
more weeks of lectures this week er is which is seven eight and nine we've got 
two lectures each week in those weeks there are no lectures after 
week nine in this course and we're going to be finishing off with probability 
distributions today and we're going to move on to the applications er in in the 
five remaining lectures so this is where we are at the moment right when we 
finished er last week we were talking about the topic of er covariance joint 
distributions for random variables and talking about working out the 
relationship between variables and this was the example that i was using it's 
all in your notes but i've written it out again to illustrate and what we have 
here is a joint distribution if you look at the if you look at the table we 
have two variables one which was related to the level of demand and one which 
related to the number of advertisements and the story was that a company was 
placing weekly advertisements and it was generating demand as a result of those 
and we had an experimental probability distribution where we've got 
probabilities in blue attached to each pair of possible 
values for the random variable and then we said well given that you've got 
these blue probabilities which we call joint probabilities you could also work 
out marginal probabilities if you know all the probabilities associated with X-
equals-nought you can add them up to find the overall probability that X-equals-
zero and that was a column sum and we've moved on to thinking not just about 
things relating to marginal distribution such as expected values or variances 
but on to the relationships between variables where we needed to look at these 
joint probabilities so this was our basic data basic example that's driving the 
subject of the lecture and we finished last week by looking at the formula for 
the covariance between random variables just cover that up for the moment and 
it looked fairly messy but we dealt with some motivation for it and remember 
that this is covariance for random variables there's a distinction between what 
goes on for random variables 
which is a theoretical kind of concept and what goes on for real data but the 
story here is we are looking at effectively average deviations of each variable 
from its mean but particularly how they do that together whether when X is 
below its mean Y tends to be or not we're looking at the average of these 
things in the sense that we're looking at how important these joint deviations 
are as measured by how likely those occurrences are corresponding occurrences 
are so we take a weighted average of these paired deviations and this double 
sum that you see here simply means that you have to add across all possible 
pairs of values of the two random variables so this was the story so the double 
sum there across X and across Y simply means that what you're doing is looking 
at all possible pairs of values of the random variable s concerned and what 
we've got is a weighted average of these paired deviations from the mean the 
weighted average is obtained by multiplying 
each paired deviation by the probability of getting that deviation and all 
those probabilities lie between nought and one so we call it a weighted average 
and that's covariance for random variables and of course if you want to 
calculate it as we're about to see it can be rather tedious but the main thing 
is to get in your ideas the motivations for what we're doing we're looking at 
the relationship between variables by looking at how likely the two variables 
seem to be both below or above their means together or opposite sides of their 
means together the E is the expected value that we discussed last week that's 
the weighted average of possible values of the random variable using 
probabilities as the weights so that's covariance for random variables if you 
want to compare it with what goes on with real data in the assessed exercise 
you've got to calculate some correlations er for which you'll need covariances 
just to compare if you were looking at covariance 
for real data here's what you would do and it's possible just to identify the 
different points in the formulae er so that you can see where the points of 
comparison occur so first of all we're not using random variables we're using 
real data and our real data when we look at covariance occurs in pairs pairs of 
values for two variables and when we've got the real data formula again we've 
got deviations of er the X variable from its sample mean and the Y variable 
from its sample mean instead of expected value and instead of a probability 
weighting the weighted average we've got a real average although use N-minus-
one instead of N so we're averaging the cross-product of the deviations from 
the mean just as we're doing with the random variable case you only see one sum 
here because this is enough to do the summing across all pairs of values 
because the random variables come in the data come in pairs a value for the X a 
value for the Y but you can see it in 
the same way we're summing across all possible pairs and these are actually the 
weights every time we put in something into here we give it a weight of one 
over N-minus-one it's the same sort of story but it's for real data rather than 
for random variables so covariance for random variables then is a rather messy 
looking thing but it's doing basically the same thing summing across all 
possible pairs comparing deviations from the means of the two random variables 
and weighting the lot taking an average so it's the same story but as with real 
data the problem with the covariance measure to capture the relationship 
between random variables is that the units of the thing a-, whether it's big or 
small will essentially depend on whether the n-, units of measurement of X and 
Y are big or small so it won't do for comparing the strength of relationships 
between different pairs of random variables because its size will change with 
the size of the random variables it's 
not a standardized measure and we know what we do with real data we take the 
covariance and we divide by the standard deviations of the two variables 
involved that's what we do with sample data and it's exactly what we do with 
random variables so we have covariance which doesn't necessarily work for us in 
the sense that it's not properly scaled and instead of covariance we use 
correlation and the correlation story is related to the covariance story in 
just the same way as it is for real data that's to say the correlation is the 
covariance divided by the two can see a mistake in the slide divided by the two 
standard deviations that should be the standard deviation of Y and a standard 
deviation definition of course is the standard deviation definition that we use 
for random variables it's the weighted average of the deviation of the random 
variable of its mean squared where the weights to the probabilities for that 
random variable alone they're the 
marginal probabilities if you don't take the square root it's the variance to 
get the standard deviation you must take the square root but the relationship 
is the same as before correlation is covariance divided by the two standard 
deviations and that gives us a measure that must lie between minus-one and plus-
one and just with the product moment correlation coefficient minus-one or plus-
one indicates a perfect linear relationship between the random variables 
something in between is weaker the closer it is to zero the weaker is the 
relationship so what we want to do in er in this lecture is just run through 
some calculations of covariance and hence of correlation and finish off with 
some more general stories about er expected value and variances so we're now 
moving into section four-point-five of the notes if you look in section four-
point-five and thereabouts you'll see the details of the calculations that i'm 
going to go through with you now so it's a sensible place to be looking 
so first of all the numbers that we're actually going to be feeding into this 
calculation are the numbers from this joint distribution for demand and 
advertising so all the information that we can possibly have about these two 
random variables is encapsulated in the joint distribution we're then going to 
summarize this joint distribution by looking at the covariance and the 
correlation between these random variables in order to do that we are going to 
have to look at the indi-, the properties individually of X and Y 'cause we're 
going to need want to know what their expected values are and we're going to 
need to know what the variances are so we can get the standard deviations so we 
are going to need the so-called marginal distributions the probabilities 
associated with specific values of each variable alone and that's what you've 
got in the column and the row sums which are illustrated in the notes so we're 
going to need the joint probabilities to 
work out the covariance and hence the correlation but on the way we're going to 
need the m-, so-called marginal probabilities the probabilities associated with 
values of the r-, each random variable taken individually 'cause we're going to 
look at deviations from the mean of each random variable so this is the basic 
information that we're going to use so to get at the covariance here's the 
covariance formula to get at the covariances we're going to need the expected 
value of X and the expected value of Y and then to get at the correlations 
we're also going to need the standard deviation of X and the standard deviation 
of Y so we o-, to get those individual quantities the expected value and the 
standard deviation we need the individual distribution the margin-, so-called 
marginal distributions so here's the s-, here's the story of the calculations 
made out in tabular form so this is all to do with our advertising demand 
example so here's the 
story on the marginal distribution of X just looking at X alone there are two 
things we want to know about X alone we want to know what its expected value is 
and we want to know what its variance is so we can take its square root to get 
its standard deviation to use in the correlation so here we've got the value 
possible values of the random variable capital-X which we denote by little-X 
and we get nought one and two from the joint distribution we were able to 
calculate the marginal probabilities associated with each of those values of X 
and there they are to calculate the expected value it is the weighted average 
of the possible values of the random variable where the weights are the 
probabilities that is to say it's each X times its probability multiplied 
together and then those individual products summed that will be the expected 
value so the expected value of X here is one-point-one-one and that's for the 
fine that gives us one thing that we need for the about 
the marginal distribution about the individual X distribution the other thing 
we need is the variance and the contributions to the variance are the 
deviations that of the random variable from its expected value how far away is 
does it get from its expected value how spread out is the distribution and in 
the end what is the average of those sorts of that sort of spread so what we 
look at is the difference between each individual value and the measure of 
central tendency the expectation we square that and we take the weighted 
average of these numbers where the weights again are the probabilities and this 
squared deviation is much more likely because it has a probability of point-
five-one than this squared deviation which only has a probability of point-one-
nine so again we've got to multiply each element in this column by its 
corresponding element in this column and add the things together which is what 
we see there in the final column except you 
don't see it okay so in here point-two-three-four-one is this number multiplied 
by this one and then the variance is just the sum of those things so as long as 
there aren't too many numbers it's er tedious but not to the point of 
exhaustion and that's the variance that's the weighted average in terms of the 
random variable the way that the random variable deviates around its mean how 
spread out the distribution of the random variable is just remember that when 
we finally get to feeding this into the formula for correlation we'll want the 
square root of this quantity so although the variance is about point-four-eight 
the standard deviation is going to be bigger than that the square root of the 
number that's less than one is bigger than the original number so it's about 
point-seven and you have to repeat these this sequence of calculations for the 
Y variable so you want the same sorts of ranges of columns for the Y variable 
and to complete the calculations 
so you need to know the values that the random variable can take the 
probabilities that it can take that value that eventually gives you the 
expected value hence you can calculate the square of the deviation of the 
values from the expected value weight those by the probabilities again add them 
up that gives you the variance the square root of which is the standard 
deviation so in black there you have the key quantities that you need to obtain 
from the marginal distributions they themselves of course are coming from the 
joint distributions everything is in the joint distribution the joint 
distribution is everything what we're doing is calculating some summary 
descriptors of that distribution both individually expected value and variance 
and eventually in terms of the relationship between the variables so those are 
the numbers that we need these calculations are laid out but we've just been 
through one to make it clear it's going so we now need to 
turn back to the joint distribution to think about these the way that the 
variables change together rather than individually okay this table is also in 
your notes it comes out rather small here but what we need to do is to start 
thinking about the relationship between the random variables so we've got to to 
move on now to the joint distribution so the first thing to think about is how 
to organize the pairs of values that can occur well if you just look at the 
first and the third column here you can see how i've organized it i've tried to 
organize it systematically i've said fix the Y value at one work through the 
possible X values fix the vy-, Y value at two work through the corr-, possible 
X values fix the Y value at three work through the possible X values just do it 
systematically you could lay it out differently but we have got to cover all 
possible pairs of values so every row in this table corresponds to a pair of 
values of X and Y so when we're trying 
to sum across all possible pairs what we're doing again is summing the column 
so in the middle column there two i've just u-, i've put the deviation of X 
from its mean why do i need that because that term feeds in to my covariance 
formula not on its own but it's still in there so here i've got all my 
deviations i'd already calculated these numbers for the purposes of calculating 
the variance of X so it's not although i've written them in again w-, you would 
have already calculated them and of course it repeats so as an an as X-equals-
nought the deviation is always minus-one-point-one-one at X-equals-two the 
deviation is always nought-point-eight-nine so it reappears and the same will 
be true of the way Y deviates it from its mean only because of the way i've got 
it organized i've got three Y-equals-one values so i've got three deviations 
the same three two values together so i'll get three values the same there but 
again i i would have already 
calculated these deviations to get at the variance i'm going to use them again 
to get at the covariance but to get at the covariance i'm going to multiply 
these elements from these two columns together i'm interested in the 
relationship between the deviation of X from its mean and Y from its mean that 
was what the formula said 
so i need a fifth column which would be a brand new column which is the product 
row by row of what's in the second column and what's in the fourth column so my 
one-point-o-four here is minus-one-point-one-one multiplied by minus-nought-
point-nine-four both numbers tend to be er both numbers there are below their 
mean next time we're multiplying minus-point-one-one by minus-point-nine-four 
again both numbers tend to be below their mean that's evidence for positive 
covariance they vary together however the next number point-eight-nine times 
minus-point-nine-four X is above its mean but Y is below [sniff] how important 
is that well we it depends 
on the probability of getting that pair of values so we've got two pairs of 
values that are both below their mean and they both have positive probabilities 
one where one's above one's below and it has a positive probability but it's 
relatively small so on balance when we look at that it would appear that we've 
got er favour for a positive covariance or evidence for positive covariance and 
then we work our way down that's below that's above multiply them together get 
a positive number again that's actually sorry we get a n-, we get the negative 
number here i've moved this across to the probability column now we're getting 
one below one above they're opposite opposite locations with respect to the 
mean and so on through so that's this column here this column here is a 
deviation of X from its mean times the deviation of Y from its mean for each 
pair of values we want to find out what the average de-, average combination of 
these deviations is when multiplied 
together so we're going to multiply each product by its probability of 
occurring that's the joint probability of getting the underlying values of X 
and Y that will then give us eventually when we add all those things together 
it'll give us a weighted average so what we want to do is multiply this column 
which itself is the product of these two by the probability and that's the 
final column that we've got in that table so the final column there is the 
element in column two times the element in column four times the element in 
column six but that's for each so we've got one row corresponds to one pair of 
values of the random variables we want to sum across all pairs we were 
expressed that as a double sum but all you have to think of is summing across 
all possible combinations so what we have to do is sum up this column of num-, 
numbers here and find out what we get and that will be the covariance between 
these two random variables now the that's just a matter 
of hitting the buttons on a calculator then it's about nought-point-one we 
can't read anything at all into the numerical value of that number because the 
number that you get out of that is clearly going to depend on the sizes of the 
numbers that went in in the first place and if these sizes are large then that 
number is going to tend to be large what we can do however is take something 
from the sign the sign is positive so whatever kind of relationship there is 
here or linear relationship of course specifically we are talking about is 
positive whatever there is it would represent a situation where if X is above 
its mean Y would tend to be above its mean if w-, X is below its mean Y would 
tend to be below its mean whether they're below or above or not they're paired 
in that sense so what we've done then is to compute the formula at the bottom 
of the screen we've taken all the possible pairs of X-minus-E-of-X Y-minus-E-of-
Y multiplied them together multiplied 
them by their probabilities and added the lot together and that's the 
covariance however we want the correlation we want this scaled version that's 
going to tell us is that er a very strong relationship or isn't it we've got to 
remove the size effect from the er measure and we do that by dividing by the 
standard deviations we've worked out what the standard deviations are on the 
last slide but one so all we have to do is to substitute them into the formula 
all the bits of the formula are now calculated all we got to do is substitute 
them so the correlation is the number we worked out by the covariance divided 
by both the standard deviation of X and the standard deviation of Y and that 
will give us a standardized measure that must lie between minus-one and plus-
one the closer it is to zero the weaker is the relationship the closer it is to 
minus-one the closer it is to a perfect downward sloping relationship the 
closer it is to plus-one the closer it 
is to a perfect upward sloping relationship so there are the numbers that we've 
calculated on the previous slides they simply have to be substituted into the 
formula you can multiply these two together first before dividing them into 
that one if you prefer or can do it sequentially divide covariance by standard 
deviation of X and then divide that number by the standard deviation of Y it's 
the same result will occur and the bottom line for the correlation between 
these two random variables is that we get a number that's about nought-point-
two that's rounded to four decimal places the number that you see up there so 
the correlation is positive but it's weak er there doesn't seem to be a strong 
relationship between er the advertising and the demand levels over the period 
that this empirical probability distribution has been constructed we shouldn't 
perhaps be too surprised about that but this number for some distributions this 
number will be a lot bigger 
but we've got now a number of summary statistics for our joint distribution 
we've got b-, in particular we got individually for the Ys and the Xs their 
expected value which is a measure of the s-, of the er central tendency or the 
general location of the distribution of each individually we've got the 
variance which is a measure of their spread about that measure of central 
tendency how spread out are they about that standard that er overall measure of 
location of the distribution and then we've got two measures of the 
relationship between the variables we've got the covariance and then we've got 
the correlation so the numbers are are er er somewhat tedious but what i'd 
really like you to take away from this is er you are you do have for the 
purposes of examinations to be able to perform such calculations but in the 
longer term what you need to develop is some intuition about what's going on 
with these definitions what it what is really 
being delivered to you when you make these calculations er because in practice 
if you do upgrade any much more statistics of course you won't be handling the 
individual calculations you'll leave that to a machine but if you don't know 
what the formula is doing you really can't interpret the answers very reliably 
so that's joint distributions then and summary statistics for distributions of 
random variables and each one of those has an analogue with real data we start 
off with real data get an understanding of what we're doing with real data and 
then to develop more statistical techniques we develop tools random variables 
and probability distributions which allow us to become more sophisticated in 
our analysis of real data there's a feedback effect 
the last er few things that i want to talk about in this section of the 
lectures er harp back to looking at individual variables looking at individual 
random variables and i want to talk then again about expected value and 
variance but i want to point out to you a very important feature of these er 
calculations very often you'll have some information about some variable that's 
not itself directly of interest the thing that you want to think about is is 
related to the er that random variable about which you have information but is 
not the same as that random variable so you might have a probability 
distribution about s-, about in this case it's going to be a about sales in 
this 
illustrative example i'm going to introduce but you may not want to know about 
sales you may not want to know what the expected value of sales is you may not 
want to know what the variance of sales is what you may want to know is what 
happens to profits what's the properties of profits not what are the properties 
of sales but i-, what you know about is sales how can you move from one to the 
other so here's a story that is very useful and this story also applies 
although we're going to introduce it for random variables and talk about 
expected values and variances exactly the same rules apply for real data in 
other words to averages and to sample variances so supposing we've got two 
we've got a random variable we've got information on sales but what we want to 
know is profits we want to know about profits so here's the relationship that 
has been found to exist between profits capital-P and sales capital-X i'm using 
capital letters because i want you at this stage to 
think about these things as random variables in general they will have 
probability distributions we won't we will be able to get realizations or 
particular values in practice but when we're thinking about them as random 
variables we want to think about them in the abstract so this says that 
whatever value of X you happen to have from your distribution i-, you've the 
corresponding value of P can be calculated and so it can be stated as a general 
rule as the relationship between random variables the units here are that er 
everything's been made in er is measured in thousands of pounds er and in some 
cases thousands of pounds per day and you can interpret the u-, the terms that 
you see here it's a linear equation it's simple and the rules that i'm going to 
deal with are specific to linear equations linear equations are ones where one 
variable is a constant multiplied by the other with another constant added or 
subtracted you've seen that before with what namex did 
for elasticity and so on and you can interpret these coefficients as i have 
done there the three is giving you some measure of the profit per car in 
thousands of pounds the minus-two is some fixed costs per day in thousands of 
pounds so what you've got here are the amount of profits that you're going to 
get per car minus the amount of money that you're going to lose anyway as a 
result of say keeping your showroom going or something of that sort so this 
relationship is in terms of random variables so R-Vs here means i'm using 
capitals i'm talking about random variables in general but i can of course use 
the rule for specific values of the random variables so er if we know or 
somebody wants asked wants to ask wants er us to consider what happens when the 
random variable X takes the value specifically one then we can work out what 
the corresponding profits would be by simply feeding that one into the formula 
and in this case we'd get profits of one unit thousand pounds per 
day we've got i-, either because we s-, er developed a statistical model or 
we've got an experimental probability distribution we've got information about 
X but what we want to know about is not X it's profits this is what matters or 
at least for some reason this is what we're asked to investigate so how do we 
get a story about profits from the story about sales and how that in terms of 
probability distributions and their summary statistics well when you look at 
the equation you can see that any value of X generates a particular value of P 
so if we change the value of X we'll necessarily change the value of P and if 
we never come back to the same value of X we'll never come back to the same 
value of P a particular value of profits of er sales X is associated uniquely 
with a particular value of profits P so if somebody tells us what the 
probability of getting some particular value of sales is all we have to do is 
to u-, to describe the probability of getting the 
corresponding value of profits is saying it's the same so the probability of 
getting profits of unit one is the same s-, as the probability of getting sales 
of unit one the probabilities are going to be the same it's called a one to one 
relationship you've got no overlapping at all so what this tells us is we can e-
, if we know what the probability distribution of the Xs is we know what the 
probability distribution of the Ps are we can just read off what the 
probability of the corresponding X value is and we'll have a set of pairs of 
values of profits with their probabilities so we'll know what the probability 
distribution of profits is from which we can calculate any of the summary 
statistics for the distribution er that we want expected value or variance 
using the formulae we've got so we can regenerate a new probability 
distribution and we can calculate expected values and variances if we want to 
know the probability distribution 
specifically we're going to have to go through this process but it turns out 
that if you want to know just these summary statistics you don't have to go 
through the irksome process of calculating all the probabilities and then 
running through the formulae there's a sh-, a very important short cut so just 
to be clear supposing you wanted to work out what the er expected value of 
profits was if you do it longhand what have you got to do well for each 
possible value of X you have to work out the corresponding value of profits and 
you have to know what the probability of getting that value of X is it'll be 
the same as the probability of getting that value of profits but you have to do 
this calculation for every single value of X you have to work out a new value 
of P to get the probability and then you're going to have to multiply the value 
of P by its probability for each possible value and add them all up to get the 
expected value er there isn't an 
answer here i haven't written it out that's a tedious calculation and if there 
were very many possible values for the random variable not just six maybe sixty 
it would be a real pain but this is how you'd work out the expected value of P 
if you had to it's minus-two times point-one-eight plus one times point-three-
nine plus four times point-two-four all the way to the end thirteen times point-
o-one and er the variance would be even worse you'd have to work out all the s-,
deviations square them multiply the probabilities add them a real pain so you 
don't do that when the relationship between the random variables is linear when 
the relationship between the random variables is linear you've got a very 
simple alternative route to follow 
so we just have a general statement and we say here's a linear relationship 
between a random variable X and a random variable Y we multiply by B add A so 
we've got all the information we need about Y we've perhaps got its probability 
distribution and we want to find out what the expected value of X is how do we 
go about it well all we have to know is that the relationship between the 
expected values is exactly the same as the relationship between the random 
variables themselves so if in order to get any value of Y you must multiply X 
by B and add A it's true that if you want the expected value of Y you simply 
multiply by B and add A so you certainly don't go through all the rigmarole of 
working out all the probabilities and expected values and the and so on all you 
need to know is 
what the expected value of X is so if somebody's told you what the expected 
value of X is or you've got that information from somewhere it's very easy to 
work out the expected value of Y the same is true of the variance but don't 
forget with the variance you're always looking at deviations from the mean and 
you're squaring them so when we look at the mean of Y or the ad-, expected 
value of Y no matter which expected value no matter what the actual value of Y 
is the the specific value of Y is the expected value is going to involve the 
term A so when we look at the difference between the average and the actual 
both the average and the actual will include a term in A here's the A coming in 
from the average here's the A coming in from the actual if you like so when we 
take the difference the A is going to disappear so when we look at deviations 
of Y about its mean clearly the A bit's not going to play a role what is going 
to play a role is the B bit but that's going to 
get squared up because variance we look at the square of the deviation of the 
mean so when we look at say this minus this the A will disappear but we'll have 
B in there when we look at variance we square that so we're going to end up 
with B-squared and in fact the relationship between the variances is this the 
variance of Y is B-squared times the variance of X you have to the A is 
irrelevant it disappears when taking the difference the B sticks around but it 
has to be squared up because variance is the squaring operation and these 
relationships would hold as well for sample data if you knew what this samp-, 
if you got some data and you knew what the average value of X was and you knew 
that X was related to Y in this way then the average value of Y could be worked 
out using this formula but plugging in the average value of X similarly if you 
knew what the sample variance of X was then the sample variance of Y can be 
worked out like this it's true both of the 
random variable case which we sometimes refer to as the population case and 
real data it's also the case that these relationships hold not just for 
discrete random variables we've been pushing the discrete random variable story 
for reasons of simplicity these relationships also hold for continuous random 
variables although we haven't defined exactly what we mean by expected value 
and variance you should be developing some conceptual ideas of what we mean and 
we are able to define these things for continuous random variables in which 
case these relationships continue to hold it's very general result when the 
basic relationship between the random variables is linear so here is er a very 
simple er illustration then we had that profits was three times sales minus two 
er once you know that the expected value of sales is one-point-five you can 
easily calculate the expected value of profits by feeding it into the formula 
you don't have to go through all the 
rigmarole of working out a probability distribution and er going through y-, 
every step in the calculation of expectation similarly with variance the A bit 
is our minus-two here it's the constant that you add on in this case it's a 
subtraction so it's minus-two but it's irrelevant and the only thing that's 
important is the multiplicative factor of three and that of course has to be 
squared so we get the variance of P is three-squared times the variance of X 
variance of X one-point-two-five and so we get the variance of P which is large 
of course 'cause the multiplicative term is larger than one we'd have got 
variance reduction if this number instead of three was a number less than one 
just depends on what's er being fed in so when there's a linear relationship 
between your random variables don't recalculate expected mean and variance use 
these formulae we make far less play of the following results but nonetheless 
they're interesting 
they're important to know about at least to be aware of there are all kinds of 
other relationships like this that exist if we want to if we've got to add 
random variables together possibly multiplied by their own constants there are 
all kinds of simple rules for working out the new expected values and the new 
variances from the old expected values and variances so a very simple case is 
if you want the expected value of the sum of two random variables it's just the 
sum of the expected values very straightforward indeed variance is more 
interesting when you come to think about sums because if you're dealing with a 
sum you don't just have to consider what's happening to each variable 
individually you have to think about what's happening to them together so if 
you're faced with a position of needing to work out the variance of a sum you 
have to take account of the covariance the way that they vary together so 
you'll notice we've got there's coefficients 
being squared up here but let's look at the really simple case to make the 
point if you wanted the variance of a sum the variance of a sum would be the 
sum of the variances that's fine but you also have to take account of the 
covariance the extent to which they vary together and if the covariance is 
negative you can see that this that will reduce the variance compared with 
simply this sum of the individual variances the covariance is helping to have a 
reducing effect on the aggregate variance in there if that er if the covariance 
is negative notice also then that in the special case where there is no 
covariance where there's no relationship at all between the random variables 
then indeed if we want the variance of the sum it's the sum of the variances 
that should remind you a little bit of the story about independence and 
probabilities if you want the probability of a joint event we said sometimes 
you can just multiply the two probabilities together 
but only if the two events are independent so the story here is if you want the 
variance of the sum you can take the sum of the variances but only if there's 
no relationship between the variables the covariance is zero so you then might 
ask well is it true that er zero covariance means that the random variables are 
independent can i say that if there's no covariance between the data they're 
really independent remember we defined independence quite precisely earlier on 
well unfortunately that's not the case the definition of independence is very 
precise and it only involves probabilities when we looked at covariance we 
weren't just interested in probabilities we were also interested in values that 
the random variables can take and it turns out that if you start off with 
random variables that really are independent then it's certainly true that 
there won't be any covariance between them the co-, rather to be c-, more 
careful the covariance will be zero between 
them that's that's okay but it doesn't go the other way that is to say just 
because when you feed everything into the covariance formula everything cancels 
out to give you zero covariance it doesn't mean strictly speaking that the 
random variables are independent and the reason is that what's going into the 
covariance formula is not just the probabilities joint probabilities and that 
it's values of the random variable too when we look at independence the only 
issues are the marginal probabilities the individual probabilities okay that's 
it for the lecture can i remind you please er there's the er assessed exercise 
which is due in this time next week if you've got questions about it come and 
speak to me or your class tutor er and the problem sets for discussion this 
week are the ones on expected value variance and covariance okay thanks very 
much that's it