91Ó°ÊÓ

Let \((X, Y)\) be a random point uniformly distributed on a unit disk. Show that \(\operatorname{Cov}(X, Y)=0,\) but that \(X\) and \(Y\) are not independent.

An item is present in a list of \(n\) items with probability \(p ;\) if it is present, its position in the list is uniformly distributed. A computer program searches through the list sequentially. Find the expected number of items searched through before the program terminates.

Show that if a density is symmetric about zero, its skewness is zero.

Find the moment-generating function of a Bernoulli random variable, and use it to find the mean, variance, and third moment.

Find the mgf of a geometric random variable, and use it to find the mean and the variance.

Use the mgf to show that if \(X\) follows an exponential distribution, \(c X(c>0)\) does also.

Find the distribution of a geometric sum of exponential random variables by using moment-generating functions.

Show how to find \(E(X Y)\) from the joint moment-generating function of \(X\) and \(Y\).

Find expressions for the approximate mean and variance of \(Y=g(X)\) for (a) \(g(x)=\sqrt{x},\) (b) \(g(x)=\log x,\) and \((c) g(x)=\sin ^{-1} x\).

The volume of a bubble is cstimated by measuring its diamcter and using the relationship $$V=\frac{\pi}{6} D^{3}$$ Suppose that the true diameter is \(2 \mathrm{mm}\) and that the standard deviation of the measurement of the diameter is .01 \(\mathrm{mm}\). What is the approximate standard deviation of the estimated volume?