Hw6_2700_s09 - U and V independent random variables? Why or...

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ENGRD2700 Basic Engineering Probability and Statistics Spring 2009 Homework 6: Joint Probability Distributions and Random Samples. Hand in by 1:30 pm on Friday, March 13 in the ENGRD2700 dropbox. Be sure to write your name and section number (1–6) on your homework. Make sure that your homework pages are stapled together. This homework has 10 parts. NOTE: Your answers will be graded on the clarity of your writing as well as the correctness of the answer. You must show your work to receive full credit. Any work that you submit must be your own. 1. Let X and Y be independent standard exponential random variables ( λ = 1). (a) Find P ( X 4 ,Y < 3) . (b) Find E ( X 2 Y ) . 2. A random point (X,Y) is distributed uniformly on the square with vertices (1,1), (1,-1), (-1,1), and (-1,-1). That is, the joint pdf is f ( x,y ) = 1 4 on the square. (a) Find P ( X 2 + Y 2 < 1) . (b) Find P (2 X - Y > 0) . (c) Find P ( | X + Y | ≤ 2) . 3. Let U = “the number of flips needed to get the first head” and let V = “the number of tosses needed to get two heads” in repeated tosses of a fair coin. Are
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Unformatted text preview: U and V independent random variables? Why or why not? 4. Let the joint pdf of ( X,Y ) be f ( x,y ) = 1 on { < x < 1 , x < y < x + 1 } and zero otherwise. Find the following: (a) The marginal distribution of X , E ( X ) and V ar ( X ). (b) The marginal distribution of Y , E ( Y ) and V ar ( Y ). (c) E ( XY ), Cov ( X,Y ), and ρ XY . Are X and Y independent? Why or why not? 5. Devore (7e) Text: 5.18 (p. 196) 6. Let X be the mean of a random sample of size n = 10 from a distribution with pdf f ( x ) = 6 x (1-x ) , < x < 1. (a) Find the mean of X . (b) Find the variance of X . 7. Devore (7e) Text: 5.52 (p. 218) 8. Devore (7e) Text: 5.60 (p. 222) 9. Let X be the mean of a random sample of size n = 36 from an exponential distribution with mean 4. (a) Find P (3 . 1 < X 1 < 4 . 6). (b) Approximate P (3 . 1 < X < 4 . 6). 10. Devore (7e) Text: 5.66 (p. 222)...
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This note was uploaded on 04/05/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell University (Engineering School).

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