Homework 1 due Tuesday, January 22, 2013 1. For each of the following distributions, find the expected value of X . (Note: f ( ; ) = ( ). x fx ? ) (a) 1 ( ; ) = (1 )x f x ?? ? ? ? , x = 1, 2,? ; 0 1 ?? ? (b) 2 fx x ( ; ) = ( 1) ? ? ? ? ? ? , x > 1, ? > 0 (c) 2 ( ; )= x f x xe ? ? ? ? , x > 0 , ? > 0 2. Let 1, , X ? Xn be a random sample from each of the distributions in problem 1. In each case, find the joint distribution of 1, , X ? Xn . 3. The amount of time (in hours), X , needed by a local repair shop, Rusts 'R Us, to repair a randomly selected piece of equipment is assumed to be an exponential random variable with pdf 1 0; 0 ( ; )= 0 x e x f x else ? ? ? ? ? ? ? ? ? ? ? ? and mgf ? ? 1 () 1 . M X t t ? ? ? ? (a) Explain in words what the random variable, X , represents in this scenario. Why is X considered a random variable? (b) Identify the parameter and explain in words what the parameter represents in this scenario. (c) In a random sample of 5 repair times, Rusts 'R Us observes the following values (in hours): 14, 6, 3, 4, 1.5 Explain in words what n, 1,..., , X Xn ?, and 1,..., n x x are in this situation, and, if applicable, give the numerical values of each. (d) Find the joint distribution of the random sample in part (c). List all assumptions made when finding this joint distribution and explain whether or not each is reasonable in this scenario. (e) Rusts 'R Us wants to estimate the expected amount of time needed to repair a piece of equipment. Identify the parameter Rusts 'R Us wants to estimate, and explain why the sample mean, X , is an appropriate statistic for the business to use. (f) Why is the statistic identified in part (e) considered a random variable? Give the observed value of the statistic, x, based on data provided in part (c). (g) Use MGFs to show the sampling distribution of ~ 5, . 5 X Gamma ? ? ? ? ? ? ? Show all work and provide rationale for each step. 4. Let 1, , X ? Xn be a random sample from the 2 N(, ) ? ? distribution, and let =1 1 = n i i X X n ? (a) Find E( ) X . (b) Find Var( ) X . 5. Let 1 2 ,,, YY Y ? n be a random sample from the population with pdf 1 0 <>< 1;=""> 0 ( ; )= 0 y y f y elsewhere ? ? ? ? ? ? ? ? (a) If = ln W Y i i ? , show that Wi follows an exponential distribution with mean 1/? . (b) Use moment generating functions to show that if W has an exponential distribution with mean 1/? then P W ? 2? has a 2 ? distribution with 2 degrees of freedom. 6. The opening prices per share Y1 and Y2 of two similar stocks are independent random variables, each with density function given by 1 ( 4) 2 1 4 ( )= . 2 0 y e y f y elsewhere ? ? ? ? ? ? ? ? On a given morning an investor is going to buy shares of whichever stock is less expensive. (a) Let V be the price per share the investor will pay, i.e., let V be the minimum of Y1 and 2 Y , denoted by min , . ? ? Y Y1 2 Find the CDF for VFv , . V ? ? (b) Find the probability density function, ? ?, Vf v for the price per share that the investor will pay. (c) Find the expected cost per share that the investor will pay, E V? ?. (d) Find the variance of the cost per share that the investor will pay, Var V? ?
Homework 2 due Tuesday, January 29, 2013 1. Let 1 2 ,,, YY Y ? n be a random sample from the population with pdf 1 0 <>< 1;=""> 0 ( ; )= 0 y y f y elsewhere ? ? ? ? ? ? ? ? For HW 1 you showed the following: ? = ln W Y i i ? follows an exponential distribution with mean 1/? ? P W ? 2? has a 2 ? distribution with 2 degrees of freedom Use this information to answer the questions that follow. (a) Show that =1 2 n i ??i W follows a 2 ? distribution with 2n degrees of freedom. (b) It turns out that if 2 X ? ? distribution with ? degrees of freedom, then 1 1 = 2 E X ? ? ? ? ? ? ? ? . Use this to show =1 1 1 = . 2( 1) 2 n i i E n ? W ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2. Let Xi , i = 1, 2,3, be independent with 2 Nii (, ) distributions. For each of the following situations, use the Xi s to construct a statistic with the indicated distribution. (a) Standard Normal Distribution, N(0,1) (b) chi squared with 3 degrees of freedom (c) t distribution with 2 degrees of freedom (d) F distribution with 1 and 2 degrees of freedom 3. Assume 1 2 , ,, X X X ? n are independent, normal random variables with E X? ?i i ? ? and ? ? 2 , 1,2,..., . Var X i n i i ? ? ? . Recall that the mgf of the ? ? 2 N ?,? distribtution is 2 2 2 . t t e ? ? ? (a) Find the distribution (including parameters) of 1 . n i i Y X ? ? ? (Hint: Use mgfs.) Explain in words what this result indicates and why you think it is useful. (b) Give an example of a problem where you would use the result found in part (a). Explain in detail how the result from part (a) would be used for the example you provided. (c) Give the distribution (including parameters) of the following statistics: o 1 2 3 5 X ? X o 2 2 Z1 2 ? Z where , 1, 2 i i i i X Z i ? ? ? ? ? o 2 1 2 Z ? Z where , 1, 2 i i i i X Z i ? ? ? ? ? o ? ? ? ? 2 1 2 2 3 4 Z Z Z Z ? ? where , 1, 2,3, 4 i i i i X Z i ? ? ? ? ? 4. Let 1, , X ? Xn be a random sample from a Gamma(a,ß) distribution. (a) PDFs of three different gamma distributions are given below. Compare and contrast the three parent population distributions in terms of shape, mean and variance. (b) Use MGFs to find the exact distribution of 1 1 . n i i X X n ? ? ? (c) Using the Central Limit Theorem, what is the approximate distribution of 1 1 ? n i i X X n ? ? ? (d) For each pair of values of a and ß listed below, create graphs of the exact and approximate distributions of X (found in parts b and c) when n = 5, n = 20, and n = 50. For each pair of parameter values, compare the exact and approximate distributions of X as n increases. ? a = 2 and ß = 2 ? a = 10 and ß = 0.4 (e) Comment on how the skewness of the parent population (i.e., the Gamma(a,ß) distribution) and the sample size affect the Central Limit Theorem approximation of the exact distribution of X.