Week 7 math 170E
what does the correlation coefficient must stay between
-1 ≤ p ≤ 1
What is the other way you can write the covariance
=E[XY]-µxµy
correlation coefficient
A numerical index of the degree of relationship between two variables. σxy/(σxσy)
Law of Total Probabilities for Expectation
If X,y are discrete r.v. such that E[x] exists then E[x] = E[E[X|y]] ie the expectation of X is equal to the expectation of all the conditional means
for least squares regression line what do we end up setting Y equal to
Y = µy + pσy/σx(x-µx)
Covariance
a measure of the joint variability of two random variables aka σxy = E[(x-µx)(y-µy)] this measures the spread X/Y and how much spread together
for each Y=y what do you have
an expectation E[x|y] so for each Y=y you have E[X|y]
why is E[X|y] a random variable
because it depends on the y value
Law of total probabilities of variance
if X,y are discrete R.V. then Var(X) = E[Var(X|y)]+var[E[X|y]] provided these all exists ie it is equal to the expectation of all the variances plus the variance of the expectations
If X,Y are independent what is the covariance and why
it is 0 because Cov[X,Y] = E[XY] - µxµy but if they are independent it means that E[XY] = E[X]E[Y] so end up with E[X]E[Y] - µxµy µxµy- µxµy =0
If Y=cX for some constant c>0 then what does it mean about the correlation coefficient
it is 1 (or -1 if c is negative)
what is the conditional variance of x given Y
it is E[(x-E[X|y])^2] for Var[X|y] So it is essentially the same as when you are normally calulating variance but you are using the conditional pmf for a specific y value it can also be computed using E[x^2|y]-E[X|Y]^2 just like with normal variance
For a discrete random variable what is the conditional pmf of X given Y = y This is really P(X=x|Y=y)
it is fx|y(X|Y) = f(x,y)/fy(y) only defined if fY(y) = 0 Remember this is more than one value still. It depends on the combination of X and Y
what does E[X|y] have
its own expectation E[E[X|y]] because it is its own random variable
when Cov(X,Y) > 0 what do we say about X, Y
that they are positively correlated
what does the correlation coefficient p measure
the degree to which Y is "linear" in X
what does p = 1 mean
the two are perfectly linearly correlated and positive
what does p = -1 mean
the two are perfectly negatively correlated and linear
what is E[E[X|y]]
this is ∑E[X|y]fy(y) over all the ys because it is a random variable dependent on y you multiply it by the pmf of each of the ys
What is the conditional mean of X given Y = y
µx|y = E[X|y] = ∑xfx|y(x,y) It is essentially the same thing. You take for a specific y value, and sum over that conditional pmf and the x values
what does Least Square Regression Line minimize
E[Y-(bx+a))^2] Since Y is the actual output and bx+a is like linear regression . Then we are squaring it (this is just minimizing the mean square error ???) Really minimizing so y=mx+b and getting right mx+b
Which part of a conditional pmf satisfies the properties of a normal pmf
It is when we fix a value Y=y then fx|y(x,y) IE all of the x values for a particular y value it will satisfy this
if covariance is 0 does this imply anything
NO
does the entire of a discrete conditional pmf have to sum to 1
No
when Cov(X,Y) < 0 what do we say about X, Y
We say X,Y are negatively correlated
Least Squares Regression Line
The is the line which best fits points on a graph. It minimizes the weighted average of vertical distance between points in space and time
