Homework Problem: Finding the Covariance and the correlation

Question

• I have an idea on how to start the problem and that's with finding the expectation of X and Y and then apply the formula of Covariance but I'm not sure since it is a biased coin. Any help would be appreciated, thank you in advance!

Answer 1

You can also go directly for covariance.

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Let $Z$ have Bernoulli distribution with parameter $p$.

For $i=1,\dots,15$ let $Z_{i}$ take value $1$ if the $i$-th toss results in heads on let it take value $0$ otherwise.

Observe that the $Z_{i}$ are iid and have the same distribution as $Z$.

Then: $$\mathsf{Cov}\left(X,Y\right)=\mathsf{Cov}\left(\sum_{i=1}^{10}Z_{i},\sum_{j=6}^{15}Z_{j}\right)=\sum_{i=1}^{10}\sum_{j=6}^{15}\mathsf{Cov}\left(Z_{i},Z_{j}\right)=5\mathsf{Var}Z=5p\left(1-p\right)$$

This because the terms where $i\neq j$ take value $0$ because of independence.

Answer 2

No need to change your way because the formula of the covariance is general. The same idea can be applied here. Both $X$ and $Y$ are binomial random variables with parameters $(10,p)$. Hence $$E\{X\}=E\{Y\}=10p$$also if we define $X_1$, $Y_1$ and $Z$ as random variables representing the number of heads in the first, last and second $5$ coin tosses, clearly they become independent and $$E\{XY\}=E\{(X_1+Z)(Y_1+Z)\}=E\{X_1Y_1\}+E\{X_1Z\}+E\{Y_1Z\}+E\{Z^2\}=25p^2+25p^2+25p^2+10p^2=85p^2$$