Problem 11
Let \(X\) be a random variable with space \(\mathcal{D}\). For \(D \subset \mathcal{D}\), recall that the probability induced by \(X\) is \(P_{X}(D)=P[\\{c: X(c) \in D\\}] .\) Show that \(P_{X}(D)\) is a probability by showing the following: (a) \(P_{X}(\mathcal{D})=1\). (b) \(P_{X}(D) \geq 0\). (c) For a sequence of sets \(\left\\{D_{n}\right\\}\) in \(\mathcal{D}\), show that $$ \left\\{c: X(c) \in \cup_{n} D_{n}\right\\}=\cup_{n}\left\\{c: X(c) \in D_{n}\right\\} $$ (d) Use part (c) to show that if \(\left\\{D_{n}\right\\}\) is sequence of mutually exclusive events, then $$ P_{X}\left(\cup_{n=1}^{\infty} D_{n}\right)=\sum_{n=1}^{\infty} P_{X}\left(D_{n}\right) $$
Problem 11
A person has purchased 10 of 1000 tickets sold in a certain raffle. To determine the five prize winners, five tickets are to be drawn at random and without replacement. Compute the probability that this person wins at least one prize. Hint: First compute the probability that the person does not win a prize.
Problem 11
Suppose \(A\) and \(B\) are independent events. In expression (1.4.6) we showed that \(A^{c}\) and \(B\) are independent events. Show similarly that the following pairs of events are also independent: (a) \(A\) and \(B^{c}\) and (b) \(A^{c}\) and \(B^{c}\).
Problem 12
Compute the probability of being dealt at random and without replacement a 13 -card bridge hand consisting of: (a) 6 spades, 4 hearts, 2 diamonds, and 1 club; (b) 13 cards of the same suit.
Problem 12
Let \(X\) be a random variable. If \(m\) is a positive integer, the expectation \(E\left[(X-b)^{m}\right]\), if it exists, is called the \(m\) th moment of the distribution about the point \(b\). Let the first, second, and third moments of the distribution about the point 7 be 3,11, and 15 , respectively. Determine the mean \(\mu\) of \(X\), and then find the first, second, and third moments of the distribution about the point \(\mu\).
Problem 12
Let \(\mathcal{C}\) be the set of points interior to or on the boundary of a
cube with edge of length 1. Moreover, say that the cube is in the first octant
with one vertex at the point \((0,0,0)\) and an opposite vertex at the point
\((1,1,1)\). Let \(Q(C)=\) \(\iiint_{C} d x d y d z\)
(a) If \(C \subset \mathcal{C}\) is the set \(\\{(x, y, z): 0
Problem 12
Let \(C_{1}\) and \(C_{2}\) be independent events with \(P\left(C_{1}\right)=0.6\) and \(P\left(C_{2}\right)=0.3\). Compute (a) \(P\left(C_{1} \cap C_{2}\right)\), (b) \(P\left(C_{1} \cup C_{2}\right)\), and (c) \(P\left(C_{1} \cup C_{2}^{c}\right)\).
Problem 12
Find the cdf \(F(x)\) associated with each of the following probability density
functions. Sketch the graphs of \(f(x)\) and \(F(x)\).
(a) \(f(x)=3(1-x)^{2}, 0
Problem 12
Let \(X\) have the pdf \(f(x)=3 x^{2}, 0
Problem 13
Generalize Exercise \(1.2 .5\) to obtain $$ \left(C_{1} \cup C_{2} \cup \cdots \cup C_{k}\right)^{c}=C_{1}^{c} \cap C_{2}^{c} \cap \cdots \cap C_{k}^{c} $$ Say that \(C_{1}, C_{2}, \ldots, C_{k}\) are independent events that have respective probabilities \(p_{1}, p_{2}, \ldots, p_{k} .\) Argue that the probability of at least one of \(C_{1}, C_{2}, \ldots, C_{k}\) is equal to $$ 1-\left(1-p_{1}\right)\left(1-p_{2}\right) \cdots\left(1-p_{k}\right) $$
