Reference no: EM132441973
Review Problems -
Problem 1 - Suppose X and Y are two independent exponential random variables with respective densities given by (λ, θ > 0)
f(x) = λe-xλ for x > 0 and g(y) = θe-yθ for y > 0
(a) Show that
Pr(X < Y) = x=0∫∞f(x){1-G(x)}dx
where G(x) is the cdf of Y, evaluated at x [that is, G(x) = P(Y ≤ x)].
(b) Using the result from part (a), show that P(X < Y) = λ/(θ+λ).
(c) You install two light bulbs at the same time, a 60 watt bulb and a 100 watt bulb. The lifetime of the 60 watt bulb has an exponential density with an average lifetime 1500 hours. The 100 watt bulb also has an exponential density with average lifetime of 1000 hours. What is the probability that the 100 watt bulb will outlast the 60 watt bulb?
Problem 2 - Suppose Xi for i = 1, . . . , n are iid Exponential(λ). Note that the Exponential distribution is a special case (α = 1) of the Gamma(α > 0, λ > 0) distribution, with pdf
f(y) = λα/Γ(α) yα-1e-λy, y > 0.
(a) Show that Y = i=1∑nXi has a Gamma distribution (specify the parameters).
(b) Find E(Yk) for any value k > 0.
(c) Use the previous part to find the mean and variance of Y.
Problem 3 - Suppose Xi for i = 1, . . . , n are iid Poisson(λ)
(a) Show that Y = i=1∑nXi has a Poisson distribution (specify the parameter).
(b) Find the mean and the variance of Y.
Problem 4 - Let X > 0 be a positive r.v. with pdf f(x) and cdf F(x). The hazard rate, h(t) of the distribution is defined to be
h(t) = limΔ→0 (Pr(t < X ≤ t + Δt | X > t))/Δt.
It can be viewed as the instantaneous rate of failure of an item (e.g., a hard drive) at time t.
(a) Show that for this problem h(t) = f(t)/F-(t).
(b) If X is Exponential(λ), find the hazard function of the distribution.
(c) A positive r.v. X is said to have a Rayleigh distribution if its pdf (α > 0) is given by
f(x) = 2αxe-αx^2 x > 0.
Find the hazard rate of the Rayleigh distribution and compare it to that of the Exponential distribution.
(d) If r.v. X is the lifetime of a hard drive, which of the two distributions mentioned in this problem would better describe X (and why).
Problem 5 - (a) Suppose that X1, . . . , Xn are iid Exponential(λ). Find the distribution of Y = min{X1, ..., Xn}.
(b) Let X be a random variable having a uniform density on (0, 1), i.e., f (x) = 1 for 0 < x < 1. Find the distribution of the random variable Y = -2 ln X.
Problem 6 - Let Xt be the number of people who enter a bank by time t > 0. Suppose further that what happens in non-overlapping time intervals are independent. (For example, the number of people who enter between 1 and 2 PM is independent of the number of people who enter between 3 and 5 PM.)
Probabilities are calculated using a Poisson distribution, Xt - Xs ∼ Poisson(t - s) for times t ≥ s ≥ 0 and X0 = 0. This means that Pr[Xt - Xs = k] = ((t-s)ke-(t-s))/k! for k = 0, 1, 2, . . . and t ≥ s ≥ 0.
(a) Find Pr[X2 = k | X1 = 1] for k = 0, 1, 2, . . . .
(b) Are X1 and X2 independent? Justify your answer.
(c) Compute E(X2).
(d) Find E[X2 | X1 = 1] and compare it to E(X2).
Problem 7 - Recall the Geometric(p) distribution where X = number of flips of a coin until you get a head (H) with Pr(H) = p. The distribution is
Pr(X = x) = (1 - p)x-1p
for x = 1, 2, . . . , with mean E(X) = x=1∑∞x(1-p)x-1 p = 1/p, which can be obtained by brute force.
An easier way to find the mean is to condition on the first toss, say Y = 0 or 1 if the first toss is T or H. Show the mean is 1/p using E(X) = EE(X | Y).
Problem 8 - Roll a fair (standard) die until a 6 is obtained and let Y be the total number of rolls until a 6 is obtained. Also, let X the number of 1s obtained before a 6 is rolled.
(a) Find E(Y).
(b) Argue that E(X | Y = y) = 1/5 (y - 1). [Hint: The word "Binomial" should be in your answer.]
(c) Find E(X).
Problem 9 - As we saw in Example 0.7, it would be nice to model the earthquake count data as having a Poisson distribution, but the sample mean and variance do not match the Poisson case. But we can use mixtures:
(a) In Example 0.6, verify that E(X) = pλ1 + qλ2.
(b) In Example 0.6, verify that Var(X) = E(X) + pq(λ2 - λ1)2.
Hint: Use that fact that Var(X) = E[Var(X | Y)] + Var[E(X | Y)].
(c) How do parts (a) and (b) help alleviate the stated problem?
Attachment:- Assignment & Example Files.rar