91Ó°ÊÓ

If the coin is assumed fair, then, for \(n=2\), what are the probabilities associated with the values that \(X\) can take on?

Suppose a die is rolled twice. What are the possible values that the following random variables can take on? (a) The maximum value to appear in the two rolls. (b) The minimum value to appear in the two rolls. (c) The sum of the two rolls. (d) The value of the first roll minus the value of the second roll.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be independent random variables, each having a uniform distribution over \((0,1)\). Let \(M=\) maximum \(\left(X_{1}, X_{2}, \ldots, X_{n}\right)\). Show that the distribution function of \(M, F_{M}(\cdot)\), is given by $$ F_{M}(x)=x^{n}, \quad 0 \leq x \leq 1 $$ What is the probability density function of \(M ?\)

An urn contains \(n+m\) balls, of which \(n\) are red and \(m\) are black. They are withdrawn from the urn, one at a time and without replacement. Let \(X\) be the number of red balls removed before the first black ball is chosen. We are interested in determining \(E[X]\). To obtain this quantity, number the red balls from 1 to \(n\). Now define the random variables \(X_{i}, i=1, \ldots, n\), by \(X_{i}=\left\\{\begin{array}{ll}1, & \text { if red ball } i \text { is taken before any black ball is chosen } \\ 0, & \text { otherwise }\end{array}\right.\) (a) Express \(X\) in terms of the \(X_{i}\). (b) Find \(E[X]\).

If \(X\) is a nonnegative integer valued random variable, show that (a) $$ E[X]=\sum_{n=1}^{\infty} P[X \geq n\\}=\sum_{n=0}^{\infty} P(X>n\\} $$ Hint: Define the sequence of random variables \(I_{n}, n \geq 1\), by $$ I_{n}=\left\\{\begin{array}{ll} 1, & \text { if } n \leq X \\ 0, & \text { if } n>X \end{array}\right. $$ Now express \(X\) in terms of the \(I_{n}\). (b) If \(X\) and \(Y\) are both nonnegative integer valued random variables, show that $$ E[X Y]=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} P(X \geq n, Y \geq m) $$

If \(X\) is uniform over \((0,1)\), calculate \(E\left[X^{n}\right]\) and \(\operatorname{Var}\left(X^{n}\right)\).

An urn contains \(2 n\) balls, of which \(r\) are red. The balls are randomly removed in \(n\) successive pairs. Let \(X\) denote the number of pairs in which both balls are red. (a) Find \(E[X]\). (b) Find \(\operatorname{Var}(X)\).

Calculate the moment generating function of the uniform distribution on \((0,1)\). Obtain \(E[X]\) and \(\operatorname{Var}[X]\) by differentiating.

Let \(X\) and \(W\) be the working and subsequent repair times of a certain machine. Let \(Y=X+W\) and suppose that the joint probability density of \(X\) and \(Y\) is $$ f_{X, Y}(x, y)=\lambda^{2} e^{-\lambda y}, \quad 0

Use Chebyshev's inequality to prove the weak law of large numbers. Namely, if \(X_{1}, X_{2}, \ldots\) are independent and identically distributed with mean \(\mu\) and variance \(\sigma^{2}\) then, for any \(\varepsilon>0\), $$ P\left\\{\left|\frac{X_{1}+X_{2}+\cdots+X_{n}}{n}-\mu\right|>\varepsilon\right\\} \rightarrow 0 \quad \text { as } n \rightarrow \infty $$