HW 1: Due Monday 9:30am

1. MGF's: Here we derive MGF's and make some good use of them.

(a) Let random variable X, have an Exponential distribution with pdf : f(x) = βe-xβ, x > 0. Derive mgf of X, denoted as mX(t). Be sure to note the bounds on constant, t.

(b) If random variable, Y , has same distribution as in (b), what is distribution of S = X + Y (assuming the random variables are independent)?

2. Transformations: Univariate CDF and Direct Methods.

(a) Sec 3.9 pg 174: 1 (CDF and Direct Method)
(b) Let random variable W, have a Normal distribution with pdf :
f(x) = 1√2πσe- 12σ2(w-µ)2, -∞ < x < ∞.

Derive pdf of random variable, Y =W-µσ, via CDF and direct methods.

(c) Derive pdf of random variable, U = Y2, via CDF and direct methods.

(d) What are the names of the popular distributions found in (b) and (c)?

3. The waiting time Y until delivery of a textbook ordered from Amazon.com is uniformly distributed over the interval from 1 to 5 days. And of course, we wait until the quarter begins to order our textbook so we rent a book while we wait for our shipment to arrive. The cost of this delay is given by U = 2Y 2 + 3. Find the probability density function for U .

4. Let X and Y be random variables for which the joint pdf is as follows: fX,Y (x, y) = 2(x + y), 0 ≤ x ≤ y ≤ 1. Find the pdf of Z = X + Y .

5. Let X be the time that the server in a single-server queue will spend on a particular customer, and let Y be the rate at which the server can operate. A popular model for the conditional distribution of X given Y = y is to say fX|Y =y(x) = ye-xy, x > 0.

Let Y have some unknown pdf fY (y). The joint pdf of (X, Y )is then fX|Y =y(x)fY (y). Because 1/Y can be interpreted as the average service time, Z = XY measure how quickly, compared to the average, that the customer is served. For example, Z = 1 corresponds to an average service time, while Z > 1 means that the customer took longer than average, and Z < 1 means that the customer was served more quickly
than the average customer.

Given this problem it seems Z should depend on Y . Introduce dummy variable W = Y and derive the joint distribution (W, Z). Using the new joint pdf, determine if the random variables are independent or not.