Random variables I've just come across a few questions on the net which I have no idea how to answer, can anyone point me towards the right direction? 


*

*Random variables $X$ and $Y$ are such that $X$ has mean 1 and variance 4, $Y$ has mean 2
and variance 9, and $\text{Corr}(X, Y ) = 1/3$ What is the variance of $3X − 2Y + 1$?

*In question 1 above, what is the covariance between $X + 2Y$ and $X − Y$ ?

*In question 1 above, if $Z$ is another random variable satisfying $E[3X −2Y +Z] = 0$,
what does the mean of the random variable $Z$ equal?
 A: The definition of variance and a few basic properties should be enough to answer the questions. For the third one, you may need to remember that the expected value operator is linear.
You can work out ${\rm Cov}(X, Y)$ from ${\rm Corr(X, Y)}$ given how the definition of ${\rm Corr}$.
A: Given
E(X) = 1,  σ2x = 4, E(Y) = 2, σ2y =  9, and Rxy = 1/3
Cov(X, Y) = Correlation (X,Y))*(SD (X)*SD(Y)
Cov(X, Y) = (1/3)*(2)(3) = 2
Random variables X and Y are such that X has mean 1 and variance 4, Y has mean 2 and variance 9, and Corr(X,Y)=1/3 What is the variance of 3X−2Y+1?
Variance of linear combination of random variable is given by
Var (aX+bY+c)=a2 Var(X) + b2 Var(Y) + 2ab Cov(X,Y)
Hence, 
Var (3X−2Y+1) = 3^2 Var(X) + (-2)^2 Var(Y) + 2 (3)(-2) Cov (X, Y)
= (9*4) + (4*9) – 12* 2
= 72 – 24 = 48
In question 1 above, what is the covariance between X+2Y and X−Y ?

Cov(X+2y, X - Y) = Cov(X, X) – Cov(X, Y) + 2 Cov(Y, X)  - 2 Cov (Y, Y)
As, Cov(aX+bY,cW+dZ) = acCov(X,W)+adCov(X,Z)+bcCov(Y,W)+bdCov(Y,Z)
Cov(X+2y, X - Y) = 4 + 2 - 2*9 = -10
As, Cov(X,X)=Var(X) = 4, Cov(Y,Y)=Var(Y) = 9 and Cov(X, Y) = Cov(Y, X) = 2
In question 1 above, if Z is another random variable satisfying E[3X−2Y+Z]=0, what does the mean of the random variable Z equal?

Given E[3X−2Y+Z]=0
Hence, 
3E(X) – 2E(Y) + E(Z) = 0
Or (3*1) – (2*2) + E(Z) = 0
Or E(Z) = 1
Mean/expectation of random variable Z is one.
