Covariance and normal distribution

Question

Let $X,Y ∼ N(0,1)$ be i.i.d. and let $U,V$ given by $U=aX+bY+c$ and $V=dX+eY+f$ have a bivariate normal distribution (here $a, b, c, d, e, f ∈ R$ with $ae − bd$ not equals to 0).

(a) What is $Cov(X, Y )$?

(b) What is $Cov(U, V )$?

(c) What are the marginal distributions of $U, V$ ?

Would the answer to (a) be $Cov(aX + b,cY + d) = acCov(X,Y )$?

and thus, (b) is $Cov(U,V)=acCov(X,Y)$?

but how do I proceed with (c)?

Thanks.

Answer 1

a) Independents implies zero covariance. As $X\perp Y$ then $\mathsf{Cov}(X,Y)=0$.

b) Covariance is linear: $\mathsf{Cov}(aX+bY+c,dX+eY+f) = a\,\mathsf{Cov}(X,dX+eY)+b\,\mathsf{Cov}(Y,dX+eY) \\ = ...$

c) Hint: The sum of independent normally distributed random variables, is a normally distributed random variable whose (a) mean is the sum of their means, and (b) variance is the sum of their variances.

Now if $U=aX + bY+c$, and $X,Y\mathop{\sim}^{\perp}\mathcal N(0,1)$, what is the distribution of $U$?

Similarly $V=dX+eY+f$

Answer 2

With a bit of thought, checking your text or notes, clues below and in Comments, plus maybe some exploring online, you should be able to handle all parts--and learn some important relationships about variances, covariances, and the normal distribution in the process.

Clues:

(a) Independence as noted by @Saty

(b) Linearity of Cov in both arguments as in another Comment and Answer.

(c) Both normal. Many references online, some on this site.

Here is a brief simulation in R for $a = 1,\; b = 2,\; c = 3,\; d = 4,\; e = 5,\,$ and $f = 6.$ Answers should be close enough (about two places) for you to check your answers.

 m = 10^6;  x = rnorm(m);  y = rnorm(m)  # indep standard normals
 a = 1;  b = 2;  c = 3;  d = 4;  e = 5;  f = 6
 u = a*x + b*y + c;  v = d*x + e*y + f
 mean(u);  sd(u);  mean(v);  sd(v)
 ## 3.002546  # approx E(U)
 ## 2.232268  # approx SD(U)
 ## 6.008503  # etc.
 ## 6.391546
 cov(x, y); cov(u, v)
 ## -0.002046419  # approx Cov(X, Y)
 ## 13.94967      # approx Cov(U, V)

If this is not enough, please leave another Comment with your progress, what you have tried, and exactly what is not clear.