Problem 2
Let \(p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere, be the pmf of the random variable \(X\). Find the mgf, the mean, and the variance of \(X\).
Problem 2
Find the complement \(C^{c}\) of the set \(C\) with respect to the space
\(\mathcal{C}\) if
(a) \(\mathcal{C}=\\{x: 0
Problem 2
Assume that \(P\left(A_{1} \cap A_{2} \cap A_{3}\right)>0\). Prove that $$ P\left(A_{1} \cap A_{2} \cap A_{3} \cap A_{4}\right)=P\left(A_{1}\right) P\left(A_{2} \mid A_{1}\right) P\left(A_{3} \mid A_{1} \cap A_{2}\right) P\left(A_{4} \mid A_{1} \cap A_{2} \cap A_{3}\right) . $$
Problem 2
For each of the following, find the constant \(c\) so that \(p(x)\) satisfies the condition of being a pmf of one random variable \(X\). (a) \(p(x)=c\left(\frac{2}{3}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere. (b) \(p(x)=c x, x=1,2,3,4,5,6\), zero elsewhere.
Problem 2
Let the space of the random variable \(X\) be \(\mathcal{C}=\\{x: 0
Problem 2
Let a bowl contain 10 chips of the same size and shape. One and only one of these chips is red. Continue to draw chips from the bowl, one at a time and at random and without replacement, until the red chip is drawn. (a) Find the pmf of \(X\), the number of trials needed to draw the red chip. (b) Compute \(P(X \leq 4)\).
Problem 2
Let \(X\) be a random variable such that \(P(X \leq 0)=0\) and let \(\mu=E(X)\) exist. Show that \(P(X \geq 2 \mu) \leq \frac{1}{2}\).
Problem 3
For each of the following distributions, compute \(P(\mu-2 \sigma
Problem 3
A coin is to be tossed as many times as necessary to turn up one head. Thus the elements \(c\) of the sample space \(\mathcal{C}\) are \(H, T H, T T H\), TTTH, and so forth. Let the probability set function \(P\) assign to these elements the respective probabilities \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\), and so forth. Show that \(P(\mathcal{C})=1 .\) Let \(C_{1}=\\{c:\) c is \(H, T H, T T H\), TTTH, or TTTTH \(\\}\). Compute \(P\left(C_{1}\right)\). Next, suppose that \(C_{2}=\) \(\\{c: c\) is TTTTH or TTTTTH \(\\}\). Compute \(P\left(C_{2}\right), P\left(C_{1} \cap C_{2}\right)\), and \(P\left(C_{1} \cup C_{2}\right)\).
Problem 3
List all possible arrangements of the four letters \(m, a, r\), and \(y .\) Let \(C_{1}\) be the collection of the arrangements in which \(y\) is in the last position. Let \(C_{2}\) be the collection of the arrangements in which \(m\) is in the first position. Find the union and the intersection of \(C_{1}\) and \(C_{2}\).
