HW6_Spring11_Sol

HW6_Spring11_Sol - ELEC210 Spring 2011 Homework 6 Solution...

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Unformatted text preview: ELEC210 Spring 2011 Homework 6 Solution 1. Let X and Y have joint pdf: () (a) Find ( ). (b) Find . (c) Find using part b. (d) Find . Solution(20’): (a) (5’) ( ( ) () ) ( ( ∫ ) ) ∫( ) | (b) (5’) ∫ ( ) ∫ | ( ) ( ( ) ) (c) (5’) ∫ () ∫( ∫ ) ( )| (d) (5’) ∫ ( ) ∫ | 2. A message requires N time units to be transmitted, where N is a geometric ( ) random variable with pmf A single new message arrives during a time unit with probability , and no messages arrive with probability . Let K be the number of new messages that arrive during the transmission of a single message. (a) Find and using conditional expectation. (b) Find the pmf of . Hint: ( ) ( ) ∑ () . Solution(20’): (a) (10’) [ ∑ [ ( ∑ ( ) ) ()( ) ( ( ∑ ) ) [ ( ( ( ) ( ) ( ) )( ) ( ) ) (b) (10’) ∑ ∑ ( ∑ For )( ) ( ) ( ) , ∑ ∑ ( () ( ) ∑ ( ) ( ( ( ( ) ∑ )) ) ( )( ( ( () ) )) ( ) ( ) ) 3. The number X of goals the Bulldogs score against the Flames has a geometric distribution with mean 2; the number of goals Y that the Flames score against the Bulldogs is also geometrically distributed but with mean 4. (a) Find the pmf of . Assume X and Y are independent. (b) What is the probability that the Bulldogs beat the Flames? Tie the Flames? (c) Find . Solution(20’): (a) (10’) As , When , similarly , ∑ ∑ When ( )( ) ( )( ) () , ∑ ∑ ( )( ) ( )( ) () (b) (6’) ∑ () , . (c) (4’) ∑ ( () ∑ () , ). 4. Let X and Y be independent random variables that are uniformly distributed in the interval . Find the pdf of . Solution(20’): For , ∫ ∫ ∫ ∫ ( ) ( ( ) )| ( ∫ ∫( () ∫ ( ( ) () ( ( ) ∫ ∫ ( )| ( ) ( ∫( ) ) ) () ) ) () ( { And the pdf is ) , ∫ So the cdf is ∫ ∫ () ∫ For ∫ () { 5. Let X, Y, Z have joint pdf: ( ) ( ) (a) Find k. (b) Find ( ) and ( ). (c) Find ( ) ( ) and ( ). (d) Find the mean vector and covariance matrix for ( Solution(20’): ). ) (a) (2’) ( ∫∫∫ ) ∫∫ ∫∫∫ ( ( ) ∫ ) ( ) so (b) (4’) ( ( ( ) ) ( ) ( ) ( ∫ ) ) ) ( ) ( Similarly, ( ∫ ( ) ( ) ∫( ( ) ) ) ( ) (c) (6’) () ( ∫∫ ∫ Similarly, we have ) ( () ) ( ( ∫∫ ) ) ( () ( ) ( ) ) (d) (8’) ∫ ∫ ( ) ( ( ∫ ) ∫ ) ( () Similarly, we have ( ) ) ( , , ) , ∫∫ ( ∫∫ ∫∫ ∫ ) ∫ ( ( ∫∫ ( ∫ [ ) ) ( ) [ ) ) Similarly, we have So ( ...
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