Suppose that a certain examination is to be taken by five

Suppose that X has the gamma distribution with parameters and , and c is a positive constant. Show that cX has the gamma distribution with parameters and /c.

Compute the quantile function of the exponential distribution with parameter .

Sketch the p.d.f. of the gamma distribution for each of the following pairs of values of the parameters and : (a) = 1/2 and = 1, (b) = 1 and = 1, (c) = 2 and = 1.

Determine the mode of the gamma distribution with parameters and .

Sketch the p.d.f. of the exponential distribution for each of the following values of the parameter :(a) = 1/2,(b) = 1, and (c) = 2

Suppose that X1,...,Xn form a random sample of size n from the exponential distribution with parameter . Determine the distribution of the sample mean X

Let X1, X2, X3 be a random sample from the exponential distribution with parameter . Find the probability that at least one of the random variables is greater than t, where t > 0.

Suppose that the random variables X1,...,Xk are independent and Xi has the exponential distribution with parameter i (i = 1, . . . , k). Let Y = min{X1,...,Xk}. Show that Y has the exponential distribution with parameter 1 + ... + k.

Suppose that a certain system contains three components that function independently of each other and are connected in series, as defined in Exercise 5 of Sec. 3.7, so that the system fails as soon as one of the components fails. Suppose that the length of life of the first component, measured in hours, has the exponential distribution with parameter = 0.001, the length of life of the second component has the exponential distribution with parameter = 0.003, and the length of life of the third component has the exponential distribution with parameter = 0.006. Determine the probability that the system will not fail before 100 hours.

Suppose that an electronic system contains n similar components that function independently of each other and that are connected in series so that the system fails as soon as one of the components fails. Suppose also that the length of life of each component, measured in hours, has the exponential distribution with mean . Determine the mean and the variance of the length of time until the system fails.

Suppose that n items are being tested simultaneously, the items are independent, and the length of life of each item has the exponential distribution with parameter . Determine the expected length of time until three items have failed. Hint: The required value is E(Y1 + Y2 + Y3) in the notation of Theorem 5.7.11

Consider again the electronic system described in Exercise 10, but suppose now that the system will continue to operate until two components have failed. Determine the mean and the variance of the length of time until the system fails.

Suppose that a certain examination is to be taken by five students independently of one another, and the number of minutes required by any particular student to complete the examination has the exponential distribution for which the mean is 80. Suppose that the examination begins at 9:00 a.m. Determine the probability that at least one of the students will complete the examination before 9:40 a.m.

Suppose again that the examination considered in Exercise 13 is taken by five students, and the first student to complete the examination finishes at 9:25 a.m. Determine the probability that at least one other student will complete the examination before 10:00 a.m

Suppose again that the examination considered in Exercise 13 is taken by five students. Determine the probability that no two students will complete the examination within 10 minutes of each other

It is said that a random variable X has thePareto distribution with parameters x0 and (x0 > 0 and > 0) if X has a continuous distribution for which the p.d.f. f (x|x0, ) is as follows: f (x|x0, ) = x 0 x+1 for x x0, 0 for x

Suppose that a random variable X has the normal distribution with mean and variance 2. Determine the value of E[(X )2n] for n = 1, 2,....

Consider a random variable X for which Pr(X > 0) = 1, the p.d.f. is f , and the c.d.f. is F. Consider also the function h defined as follows: h(x) = f (x) 1 F (x) for x > 0. The function h is called the failure rate or the hazard function of X. Show that if X has an exponential distribution, then the failure rate h(x) is constant for x > 0.

It is said that a random variable has the Weibull distribution with parameters a and b (a > 0 and b > 0) if X has a continuous distribution for which the p.d.f. f (x|a, b) is as follows: f (x|a, b) = b ab xb1 e(x/a)b for x > 0, 0 for x 0. Show that if X has this Weibull distribution, then the random variable Xb has the exponential distribution with parameter = ab

It is said that a random variable X has an increasing failure rate if the failure rate h(x) defined in Exercise 18 is an increasing function of x for x > 0, and it is said that X has a decreasing failure rate if h(x) is a decreasing function of x for x > 0. Suppose that X has the Weibull distribution with parameters a and b, as defined in Exercise 19. Show

Let X have the gamma distribution with parameters > 2 and > 0. a. Prove that the mean of 1/X is /( 1). b. Prove that the variance of 1/X is 2/[( 1)2 ( 2)].

Consider the Poisson process of radioactive particle hits in Example 5.7.8. Suppose that the rate of the Poisson process is unknown and has the gamma distribution with parameters and . Let X be the number of particles that strike the target during t time units. Prove that the conditional distribution of given X = x is a gamma distribution, and find the parameters of that gamma distribution.

Let F be a continuous c.d.f. satisfying F (0) = 0, and suppose that the distribution with c.d.f. F has the memoryless property (5.7.18). Define (x) = log[1 F (x)] for x > 0. a. Show that for all t, h > 0, 1 F (h) = 1 F (t + h) 1 F (t) . b. Prove that (t + h) = (t) + (h) for all t, h > 0. c. Prove that for all t > 0 and all positive integers k and m, (kt/m) = (k/m)(t). d. Prove that for all t, c > 0, (ct) = c(t). e. Prove that g(t) = (t)/t is constant for t > 0. f. Prove that F

Review the derivation of the Black-Scholes formula (5.6.18). For this exercise, assume that our stock price at time u in the future is S0eu+Wu, where Wu has the gamma distribution with parameters u and with > 1. Letr be the risk-free interest rate. a. Prove that eruE(Su) = S0 if and only if = r log(/[ 1]). b. Assume that = r log(/[ 1]). Let R be 1 minus the c.d.f. of the gamma distribution with parameters u and 1. Prove that the risk-neutral price for the option to buy one share of the stock for the price q at time u is S0R(c[ 1]) qeruR(c), where c = log q S0  + u log 1  ru. c. Find the price for the option being considered when u = 1, q = S0, r = 0.06, = 1,