Problem 4
Let \(p_{X}(x)\) be the pmf of a random variable \(X\). Find the cdf \(F(x)\) of \(X\) and sketch its graph along with that of \(p_{X}(x)\) if: (a) \(p_{X}(x)=1, x=0\), zero elsewhere. (b) \(p_{X}(x)=\frac{1}{3}, x=-1,0,1\), zero elsewhere. (c) \(p_{X}(x)=x / 15, x=1,2,3,4,5\), zero elsewhere.
Problem 4
Cast a die two independent times and let \(X\) equal the absolute value of the difference of the two resulting values (the numbers on the up sides). Find the pmf of \(X\). Hint: It is not necessary to find a formula for the pmf.
Problem 4
If the variance of the random variable \(X\) exists, show that $$ E\left(X^{2}\right) \geq[E(X)]^{2} $$
Problem 5
Let a random variable \(X\) of the continuous type have a pdf \(f(x)\) whose graph is symmetric with respect to \(x=c\). If the mean value of \(X\) exists, show that \(E(X)=c\) Hint: Show that \(E(X-c)\) equals zero by writing \(E(X-c)\) as the sum of two integrals: one from \(-\infty\) to \(c\) and the other from \(c\) to \(\infty\). In the first, let \(y=c-x\); and, in the second, \(z=x-c .\) Finally, use the symmetry condition \(f(c-y)=f(c+y)\) in the first.
Problem 5
Let the sample space be \(\mathcal{C}=\\{c: 0
Problem 5
Let \(X\) be a number selected at random from a set of numbers \(\\{51,52, \ldots, 100\\}\). Approximate \(E(1 / X)\) Hint: Find reasonable upper and lower bounds by finding integrals bounding \(E(1 / X)\).
Problem 5
Let the probability set function of the random variable \(X\) be
$$
P_{X}(C)=\int_{C} e^{-x} d x, \quad \text { where } \mathcal{C}=\\{x:
0
Problem 5
By the use of Venn diagrams, in which the space \(\mathcal{C}\) is the set of points enclosed by a rectangle containing the circles \(C_{1}, C_{2}\), and \(C_{3}\), compare the following sets. These laws are called the distributive laws. (a) \(C_{1} \cap\left(C_{2} \cup C_{3}\right)\) and \(\left(C_{1} \cap C_{2}\right) \cup\left(C_{1} \cap C_{3}\right)\). (b) \(C_{1} \cup\left(C_{2} \cap C_{3}\right)\) and \(\left(C_{1} \cup C_{2}\right) \cap\left(C_{1} \cup C_{3}\right)\).
Problem 5
Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).
Problem 5
A hand of 13 cards is to be dealt at random and without replacement from an ordinary deck of playing cards. Find the conditional probability that there are at least three kings in the hand given that the hand contains at least two kings.
