I'm trying to solve this question but I'm still a little iffy on joint distribution functions:
Let $X$ and $Y$ be two independent discrete random variables with the following Joint probability mass function
| $Y = 0$ | $Y = 1$ | |
|---|---|---|
| $X = 0$ | $\frac{1}{2}$ | $\frac{1}{4}$ |
| $X = 1$ | $\frac{1}{4}$ | 0 |
Find $\operatorname{Cov}(5X - \frac{1}{2}, Y + 4)$.
Any help on this?
I tried a few ways but none have really worked. I tried $\operatorname{Cov}(XY) = E(XY) - E(X)E(Y)$, but couldn't figure out how to fit it here. Same for the method where I have to find the mean of each. I'd really appreciate any help, would go a long way!
Indeed: $$\mathsf{Cov}(X,Y)=\mathsf E(XY)-\mathsf E(X)\,\mathsf E(Y)$$
And, since $X$ and $Y$ are Bernoulli: $$\begin{align}\mathsf E(XY)&=\sum_{x,y} xy\,\mathsf P(X\,{=}\,x,Y\,{=}\,y)\\[1ex]&=\mathsf P(X\,{=}\,1,Y\,{=}\,1)&&xy=0\text{ when either is }0\\[2ex]\mathsf E(X)&=\sum_{x,y} x\,\mathsf P(X\,{=}\,x,Y\,{=}\,y)\\[1ex]&=\mathsf P(X\,{=}\,1,Y\,{=}\,0)+\mathsf P(X\,{=}\,1,Y\,{=}\,1)\\[2ex]\mathsf E(Y)&=\sum_{x,y}y\,\mathsf P(X\,{=}\,x,Y\,{=}\,y)\\[1ex]&=\mathsf P(X\,{=}\,0,Y\,{=}\,1)+\mathsf P(X\,{=}\,1,Y\,{=}\,1)\end{align}$$
By the Bilinearity of Covariance when $a,b,c,d$ are constants and $W,X,Y,Z$ are random variables:
$$\mathsf{Cov}(aX+bW,cY+dZ)={{ac\,\mathsf{Cov}(X,Y)}+{ad\,\mathsf{Cov}(X,Z)}\\+{bc\,\mathsf{Cov}(W,Y)}+{bd\,\mathsf{Cov}(W,Z)}}$$
In this case:
$$\mathsf{Cov}(aX+b, cY+d)=ac\,\mathsf{Cov}(X,Y)$$