homework10

homework10 - c. Are X and Y independent in each of the...

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Probability for Engineering Applications Assignment #10, due at the beginning of class on Thursday, April 13 th , 2006 1. (2 pts) (Textbook exercise 4.3) Let the random variables X , Y and Z be independent random variables. Find the following probabilities in terms of F X ( x ), F Y ( y ) and F Z ( z ). a. P [ | X | < 5 , Y > 2 , Z 2 2]. b. P [ X > 5 , Y < 0 , Z = 1]. c. P [min( X, Y, Z ) > 2]. d. P [max( X, Y, Z ) < 6]. 2. (3 pts) (Textbook exercise 4.5) i. ii. iii. X Y - 1 0 1 - 1 1 6 0 1 6 0 0 1 3 0 1 1 6 0 1 6 X Y - 1 0 1 - 1 1 9 1 9 1 9 0 1 9 1 9 1 9 1 1 9 1 9 1 9 X Y - 1 0 1 - 1 0 0 1 3 0 0 1 3 0 1 1 3 0 0 a. Find the marginal pmf’s for the pairs of random variables with the indicated joint pmf. b. Find the probability of the events A = { X 0 } , B = { X Y } and C = { X = - Y } for the above joint pmf’s.
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Unformatted text preview: c. Are X and Y independent in each of the three cases? 3. (3 pts) (Textbook exercise 4.10) The random vector variable ( X, Y ) has the joint pdf f X,Y ( x, y ) = k ( x + y ) &lt; x &lt; 1 , &lt; y &lt; 1 a. Find k . b. Find the joint cdf of ( X, Y ). c. Find the marginal pdf of X and of Y . d. Are X and Y independent? 4. (2 pts) Find the joint CDF F X,Y ( x, y ) when X and Y have the joint pdf f X,Y ( x, y ) = ( 2 y x 1 otherxise 1...
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