Consider an experiment in which a fair coin is tossed 10 times in a row (the tosses are independent of each other). Let X denote the number of heads observed and let Y=X^2. Find the covariance between X and Y.
I know that Cov(x,y) = E(x*y) - E(x) * E(y) but I cannot seem to get the right answer. Any hints as to where to go from here?
Hints:
$\operatorname{cov}(X,Y) = E[XY] - E[X]E[Y] = E[X^3] - E[X]E[X^2]$ where you should know (or be able to compute)
those last two expectations in the formula. If not, look in your textbook
for binomial random variable and general properties of variance.
It is somewhat harder to find $E[X^3]$ but you could try and find
$E[X(X-1)(X-2)]$ (write out the first six terms of the formula
for $E[X(X-1)(X-2)]$ without using any binomial coefficients or factorial
signs), and then use the fact that
$$E[X(X-1)(X-2)] = E[X^3 -3X^2 + 2X] = E[X^3] - 3E[X^2] + 2E[X]$$
where you know, or have just computed, the values of three of the four
expectations, and so can easily deduce the value of the fourth.
