# set7 - i, j ∈ { 1 , 2 , 3 } elsewhere. (a) Find the value...

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ECSE-305, Winter 2009 Probability and Random Signals I Assignment #7 Posted: Tuesday, March 12, 2009. Due: Tuesday, March 19, 2009, 2h30pm. Important notes: Assignments without this cover page will be discarded. Student #1: Name: ID: Student #2: Name: ID: Question Marks 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Total

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1. Suppose that for any positive integer n , the n th moment of a random variable X , is given by E ( X n ) = ( n + 1)! 2 n . Obtain a closed form expression for ψ ( ω ), the characteristic function of X . 2. Let X be a a continuous RV with the probability density function f ( x ) = 6 x (1 - x ), if 0 x 1 and 0 elsewhere. (a) Find the characteristic function of X . (b) Using the characteristic function, ±nd E ( X ). 3. Using the moment-generating function of a poisson random variable X with parameter λ , ±nd E ( X ) and V ar ( X ). 4. Let ψ X ( ω ) = 1 / (1 + ) be the moment-generating function of a ran- dom variable X . Find the moment-generating function of the random variable Y = 2 X + 1. 5. Let the joint probability mass function of two jointly distributed dis- crete RVs X and Y be p ( i, j ) = b k ( i + j ) if
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Unformatted text preview: i, j ∈ { 1 , 2 , 3 } elsewhere. (a) Find the value of the constant k . (b) Calculate P ( X = 1 , Y < 3), P ( X = 1 , Y ≤ 3), P ( X = 2), P ( X < Y ), P ( X ≤ Y ). 6. Let the joint PMF of discrete RVs X and Y be p ( i, j ) = b k ( i 2 + j 2 ) if ( i, j ) ∈ { (1 , 1) , (1 , 3) , (2 , 3) } elsewhere. (a) Find the value of the constant k . (b) Find the marginal PMFs of X and Y . 7. The joint probability density function of random variables X and Y is given by f ( x, y ) = b 2 if 0 ≤ y ≤ x ≤ 1 0 elsewhere. (a) Calculate the marginal PDFs of X and Y . (b) Calculate P ( X < 1 / 2), P ( X < 2 Y ), and P ( X = Y ). 2 8. On a line segment AB of length l , two points C and D are placed at random and independently. What is the probability that C is closer to D than to A ? 9. Two RVs X and Y are jointly uniform on [0 , 1] 2 . Calculate the proba-bility P ( Y ≤ X and X 2 + Y 2 ≤ 1). 3...
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## This note was uploaded on 09/30/2009 for the course ECSE 305 taught by Professor Champagne during the Spring '09 term at McGill.

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set7 - i, j ∈ { 1 , 2 , 3 } elsewhere. (a) Find the value...

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