Statistics 100a Homework 5 Solutions

Unformatted text preview: Solution 3. Y = 100 , 000 X i = 1 X i , Y N y = 100 , 000 X i = 1 E ( X i ) , 2 y = 100 , 000 X i = 1 Var ( X i ) = N (100000E( X i ) , 100 , 000Var( X i )) = N (500000 , 130000) Therefore, P ( Y > 500 , 500) = P Z > 500 , 500- 500 , 000 130 , 000 ! = P ( Z > 1 . 39) = . 0823 Problem 4. David works at a customer call center. He talks to customers on the telephone. The length (in hours) of each conversation has density f ( x ) = 3 e- 3 x for x > 0. The length of calls are independent. As soon as one conversation is finished, he hangs up the phone, and immediately picks up the phone again to start another call (i.e., there are no gaps in between the calls). Thus, if he conducts n phone calls in a row, the total amount of time he spends on the telephone is X 1 + .... + X n , where the X j s are independent, and each X j has the density above. What is the probability that David can make 135 phone calls within a 40 hour period? Solution 4. Time X i exp ( = 3) , E( X i ) = 1 / 3 , Var( X i ) = 1 / 9 , i = 1 , ..., 135 E 135 X i = 1 X i = 135 E ( X i ) = 45 , Var 135 X i = 1 X i = 135 Var ( X i ) = 15 P 135 X i = 1 X i < 40 = P Z < 40- 45 15 =- 1 . 29 ! = . 09828 March 14, 2013 2 Stat 100 -Intro Probability Homework 8 (last hwk) J. Sanchez UCLA Department of Statistics Problem 5. Let X be the cosine of the angle at which electrons are emitted in muon decay. X is a random variable with the following density function. f ( x ) = 1 + X 2- 1 x 1 is a constant that must have value between- 1 and 1. (a) Find the Expected cosine of the angle and the variance. Show work. (Notice that the answer will depend on since that is a constant parameter). (b) Let X 1 , X 2 , ...., X n be n independent and identically distributed random variables with the density given above each. Consider now the following functions (i) Y = 3 n i = 1 X i n ; (ii ) W = n i = 1 X i n . Compute the expected value and variance of these two functions separately. Which one has the smallest standard deviation? Show work. Solution 5. (a) E( X ) = Z 1- 1 X 1 + X 2 dx = 3 E( X 2 ) = Z 1- 1 X 2 1 + X 2 dx = 1 3 Var( X ) = E( X 2 )- [E( X )] 2 = 3- 2 9 (b) E ( Y ) = E 3 X i n ! = 3 E ( X i ) n = 3 n 3 n = Var ( Y ) = Var 3 X i n ! = 9 Var( X i ) n 2 = 9 n (3- 2 ) 9 n 2 = 3- 2 n E ( W ) = E X i n ! = E( X i ) n = n 3 n = 3 Var ( W ) = E X i n ! = Var( X i ) n 2 = n (3- 2 ) 9 n 2 = 3- 2 9 n W = p Var ( W ) = r 3- 2 9 n < r 3- 2 n = p Var ( Y ) = Y Problem 6. A certain DJ takes requests for songs at a party. Assume that there are 120 people at the party, each of whom makes exactly one request for a song. All of their requests are made independently. Assume that each personwhom makes exactly one request for a song....
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