# Find the correlation of $2$ random variables

It is given that $X,Y,Z$ are independent random variables, where each are binomial distributed with $n=100$ and $p=0.2$. If $U=X+2Y$ and $V=Z+3X$, what is the correlation of U and V?

I attempted to solve this problem by first finding the pdf of $U$ and $V$, using the formula with the Jacobian. However, I do not know how to find the Jacobian given 3 variables already!

Guide:

The following might be useful to you.

$$\rho(U,V) = \frac{Cov(U,V)}{\sqrt{\operatorname{Var}(U)\operatorname{Var}(V)}}$$

You just have to find the covariance between $U$ and $V$ and their variance.

Note that

$$Cov(aX+bY, cW+dV) = acCov(X,W)+adCov(X,V)+bcCov(Y,W)+bdCov(Y,W)$$

To compute these variables' correlation, you need to find their covariance, as $$Cov(U,V)=E(UV)-E(U)E(V).$$

Let's find the first term.

Note that $$UV = (X+2Y)(Z+3X)=XZ+3X^2+2YZ+6XY.$$

Therefore, $$E(UV)=E(XZ)+3E(X^2)+2E(YZ)+6E(XY).$$

Because $X,Y,Z$ are independent, $$E(UV)=E(X)E(Z)+3E(X^2)+2E(Y)E(Z)+6E(X)E(Y).$$

Each one of them follows a binomial distribution with $n=100$, $p=0.2$, so $$E(X)=E(Y)=E(Z)=np=100\cdot0.2=20.$$

So $$E(UV)=20\cdot20+3E(X^2)+2\cdot20\cdot20+6\cdot20\cdot20=3600+3E(X^2).$$

We just need to find the value of $E(X^2)$ in this sum to know $E(UV)$.

Notice also that the variance of the binomial is $$Var(X)=Var(Y)=Var(Z)=np(1-p)=100\cdot0.2\cdot0.8=16.$$

And the variance formula is $$Var(X)=E(X^2)-E(X)^2,$$ so we can find $E(X^2)$ as $$E(X^2)=Var(X)+E(X)^2=16+(20)^2=416.$$

Plugging it into the previous result, $$E(UV)=3600+3E(X)=3600+3\cdot416=4848.$$

Now let's find the second term of the covariance formula.

Because the expectation is linear, $$E(U)=E(X+2Y)=E(X)+2E(Y)=20+2\cdot20=60,$$ $$E(V)=E(Z+3X)=E(Z)+3E(X)=20+3\cdot20=80,$$

so

$$E(U)E(V)=60\cdot80=4800.$$

These two parts together make $$Cov(U,V)=E(UV)-E(U)E(V)=4848-4800=48.$$

We have their covariance, and now we need their variances to find their correlation.

Remember that, if $X$ and $Y$ are independent, $$Var(aX+bY)=a^2Var(X)+b^2Var(Y),$$

so $$Var(U)=Var(X+2Y)=Var(X)+2^2Var(Y)=16+4\cdot16=80,$$ $$Var(V)=Var(Z+3X)=Var(Z)+3^2Var(X)=16+9\cdot16=160.$$

All together, $$\rho(U,V)=\frac{Cov(U,V)}{\sqrt{Var(U)Var(V)}}=\frac{48}{\sqrt{80\cdot160}}=0.42.$$