Suppose that a random variable X can take only the values

Suppose that a random variable X has the Bernoulli distribution with parameter p = 0.7. (See Definition 3.1.5.) Sketch the c.d.f. of X.

Suppose that a random variable X can take only the values 2, 0, 1, and 4, and that the probabilities of these values are as follows: Pr(X = 2) = 0.4, Pr(X = 0) = 0.1, Pr(X = 1) = 0.3, and Pr(X = 4) = 0.2. Sketch the c.d.f. of X.

Suppose that a coin is tossed repeatedly until a head is obtained for the first time, and let X denote the number of tosses that are required. Sketch the c.d.f. of X.

Suppose that the c.d.f. F of a random variable X is as sketched in Fig. 3.9. Find each of the following probabilities: a. Pr(X = 1) b. Pr(X < 0) c. Pr(X 0) d. Pr(X = 1) e. Pr(0 < X 3) f. Pr(0 5) k. Pr(X 5) l. Pr(3 X 4)

Suppose that the c.d.f. of a random variable X is as follows: F (x) = 0 for x 0, 1 9 x2 for 0 < x 3, 1 for x > 3. Find and sketch the p.d.f. of X.

Suppose that the c.d.f. of a random variable X is as follows: F (x) =  ex3 for x 3, 1 for x > 3. Find and sketch the p.d.f. of X.

Suppose, as in Exercise 7 of Sec. 3.2, that a random variable X has the uniform distribution on the interval [2, 8]. Find and sketch the c.d.f. of X.

Suppose that a point in the xy-plane is chosen at random from the interior of a circle for which the equation is x2 + y2 = 1; and suppose that the probability that the point will belong to each region inside the circle is proportional to the area of that region. Let Z denote a random variable representing the distance from the center of the circle to the point. Find and sketch the c.d.f. of Z.

Suppose that X has the uniform distribution on the interval [0, 5] and that the random variable Y is defined by Y = 0 if X 1, Y = 5 if X 3, and Y = X otherwise. Sketch the c.d.f. of Y

For the c.d.f. in Example 3.3.4, find the quantile function.

For the c.d.f. in Exercise 5, find the quantile function.

For the c.d.f. in Exercise 6, find the quantile function.

Suppose that a broker believes that the change in value X of a particular investment over the next two months has the uniform distribution on the interval [12, 24]. Find the value at risk VaR for two months at probability level 0.95.

Find the quartiles and the median of the binomial distribution with parameters n = 10 and p = 0.2.

Suppose that X has the p.d.f. f (x) =  2x if 0

Find the quantile function for the distribution in Example 3.3.1.

Prove that the quantile function F 1 of a general random variable X has the following three properties that are analogous to properties of the c.d.f.: a. F 1 is a nondecreasing function of p for 0 0 F 1(p) and x1 = limp1 p<1 F 1(p). Then x0 equals the greatest lower bound on the set of numbers c such that Pr(X c) > 0, and x1 equals the least upper bound on the set of numbers d such that Pr(X d) > 0. c. F 1 is continuous from the left; that is F 1(p) = F 1(p) for all 0

. Let X be a random variable with quantile function F 1. Assume the following three conditions: (i) F 1(p) = c for all p in the interval (p0, p1), (ii) either p0 = 0 or F 1(p0) c for p > p1. Prove that Pr(X = c) = p1 p0.

Let X be a random variable with c.d.f. F and quantile function F 1. Let x0 and x1 be as defined in Exercise 17. (Note that x0 = and/or x1 = are possible.) Prove that for all x in the open interval(x0, x1), F (x)is the largest p such that F 1(p) x.

Let X be a random variable with c.d.f. F and quantile function F 1. Let x0 and x1 be as defined in Exercise 17. (Note that x0 = and/or x1 = are possible.) Prove that for all x in the open interval(x0, x1), F (x)is the largest p such that F 1(p) x.