test2B_sol - AMS310 TEST 2 Fall 2003 FORM B 1 PRINT YOUR...

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AMS310 TEST 2 Fall 2003 FORM B 1. PRINT YOUR NAME HERE______________________________ USE ONLY UPPER CASE LETTERS. UNDERLINE YOUR LAST NAME TWICE. 2. Write your student ID number here____________________________ 3. CHECK TO MAKE SURE THAT YOUR TEST HAS 8 PAGES INCLUDING THIS ONE. 4. SHOW YOUR WORK FOR ALL QUESTIONS IN SECTION II. NO CREDIT WILL BE GIVEN FOR A NUMBER WITH NO ARGUMENT. POINTS POINTS OFF POINTS EARNED SECTION I -1 30 QUESTION II-1 20 -------------------------------------------------------------------------------------------------- QUESTION II-2 20 -------------------------------------------------------------------------------------------------- QUESTION II-3 15 -------------------------------------------------------------------------------------------------- QUESTION II-4 10 -------------------------------------------------------------------------------------------------- QUESTION II-5 5 ----------------------------------------------------------------------------------------------- TOTAL 100 - __________ = ___________(SCORE) Extra Credit 5 1
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FORM __B_ Page 2 of 8 Print your name here:______________________________ WRITE THE CORRECT ANSWER IN THE SPACE PROVIDED BY THE QUESTION NUMBER Part I questions: Full credit for correct answer. No partial credit _ 0.5 __ 1. Suppose f(x) = 1/5 for -3< X< 2 and f(x) = 0 otherwise. That is X has uniform, U (-3,2) p.d.f. Find the mean value of X. Hint: dx x 5 / = x 2 /10. Ans)) 2 2 2 3 3 1 4 9 5 0.5 5 10 10 10 10 x x dx - - � � - = = - = = - � � � � ° Suppose that Z has standard normal density (Z~N(0,1)). The p.d.f. of Z is f(z) = 2 / 2 2 1 x e - π for < < - x . Use Table 3 for the Standard Normal Distribution Function to answer Questions 2 through 5 below. _ 0.0475 __ 2. P (Z< -1.67) = _ 0.0475 ___ 3. P (Z > 1.67) =____ __ 0.25 ___ 4. z 0.40 = i.e., Find c such that P(Z> c) = 0.40. __ 5 __ 5. Suppose that X has mean μ =3 and σ =0.5 and Y= 10-10X. The standard deviation of Y equals. Ans)) ( ) (10 10 ) ( 10 ) 10 ( ) 10 (0.5) 5.0 SD Y SD X SD X SD X = - = - = = = _____ _ 168 ___ 6. Suppose that X 1 , X 2 and X 3 have mean μ =10 and σ 2 = 2 and Cov(X j , X k ) =0 for j k . Then the variance of Y= 8X 1 –4X 2 – 2X 3 +2 equals_______ Ans)) 1 2 3 1 2 3 ( ) (8 4 2 2) 64 ( ) 16 ( ) 4 ( ) Var Y Var X X X Var X Var X Var X = - - + = + + (64 16 4) 2 84 2 168 = + + = = 2
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Form_B_ Page 3 of 8 PRINT YOUR NAME HERE_________________________ Part I questions: Full credit for correct answer. No partial credit 7-8. Suppose X is a continuous random variable with normal p.d.f. Suppose also that the mean value of X equals 1 and the standard deviation of X equals 4, i.e. μ =1 and σ =4 __ -1.25 __ 7. Pr{ X < -4}= P(Z< ____) where Z has standard normal distribution. Ans)) 4 1 5 ( 4) ( ) ( ) 4 4 P X P Z P Z - - < - = < = < - __ -4.12 __ 8. Find the 10 th percentile for X; i.e., find c such that P(X<c) = 0. 10.
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