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Stat 366 Lab 6 Problems (October 19, 2006) page 1 TA: Yury Petrachenko, CAB 484, [email protected], http://www.ualberta.ca/∼yuryp/
The Method of Distribution Functions
6.8 The total time from arrival to completion of service at a fast-food outlet, Y 1 , and the time spent waiting in line before arriving at the service window, Y 2 , are given with the joint density function f (y 1 , y 2 ) =
e−y^1 , 0 ≤ y 2 ≤ y 1 < ∞, 0 , elsewhere. Another random variable of interest is U = Y 1 − Y 2 , the time spent at the service window.
(a) Find the probability density function for U. (b) Find E[U ] and V [U ].
6.9 Suppose that two electronic components in the guidance system for a missile operate inde- pendently and that each has a length of life governed by the exponential distribution with mean 1 (with measurements in hundreds of hours). (a) Find the probability density function for the average length of life of the two components. (b) Find the mean and variance of this average.
The Method of Transformations
6.21 Continuing Exercise 6.9. Use the method of transformations to obtain the density function for the average life length of the two components.
6.31 Let Y 1 and Y 2 be independent random variables, both uniformly distributed on (0,1). Find the probability density function for U = Y 1 Y 2.
Stat 366 Lab 6 Problems (October 19, 2006) page 2 The Method of Moment-Generating Functions
6.38 The weight (in pounds) of “medium-size” watermelons is normally distributed with mean 15 and variance 4. A packing container for several melons has a nominal capacity of 140 pounds. What is the maximum number of melons that should be placed in a single packing container if the nominal weight limit is to be exceeded only 5% of the time?
6.50 The negative binomial random variable Y can be written as Y = ∑ri=1 Wi, where W 1 , W 2 ,
... , Wr are independent geometric random variables with parameter p. (a) Use this fact to derive the moment-generating function for Y. (b) Use the moment-generating function to show that E[Y ] = r/p and V [Y ] = r(1 − p)/p^2. (c) Find the conditional probability function for W 1 given that Y = W 1 +W 2 +· · ·+Wr = m.
Order Statistics
6.58 Let Y 1 and Y 2 be independent and uniformly distributed over the interval (0,1).
(a) Find the probability density function of U 1 = min(Y 1 , Y 2 ). (b) Find E[U 1 ] and V [U 1 ].
6.67 The opening prices per share Y 1 and Y 2 of two similar stocks are independent random variables, each with a density function given by
f (y) =
2 e
− y− (^2 4) , y ≥ 4 ,
0 , elsewhere.
On a given morning an investor is going to buy shares of whichever stock is less expensive.
(a) Find the probability density function for the price per share that the investor will pay. (b) Find the expected cost per share that the investor will pay.
