# HW5 - = | ~ ~ y y z f z 4 Let ~ x and ~ y be...

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Middle East Technical University Electrical and Electronics Engineering Department EE230 Homework 5 Due : Apr. 28, 2006 1. Let the random variable ~ x be uniformly distributed over (0,3) and the function be defined as follows: ( ) - = x x x x x x x g 2 , 1 1 2 1 , 1 1 0 , 0 , 0 Find and sketch the distribution function and the pdf of ) ( ~ ~ x g y = . 2. Let the joint pdf of ~ x and ~ y be given as < < < < = elsewhere 0 1 0 and 1 0 for 2 ) , ( ~ ~ -x y x y x f y x Let the random variable ) , min( ~ ~ ~ y x z = . Find and sketch the distribution function and the pdf of ~ z . Hint : It would be useful to consider the complement of the event } ~ { z z . 3. Let ~ x and ~ y be independent random variables having the following pdfs: ( ) ( ) ( ) ( ) 2 / 2 ~ ~ 2 1 5 3 1 3 3 2 y y x e y f x x x f - π = + δ + - δ = Let ~ ~ ~ y x z + = . a) Find and sketch ( ) z f z ~ . b) Find and sketch = } 3 { | ~ ~ x z f z , - = } 5 { | ~ ~ x z f z , and

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Unformatted text preview: = } { | ~ ~ y y z f z . 4. Let ~ x and ~ y be i.i.d. (independent and identically distributed) random variables and exponentially distributed with parameter 1, i.e., ( ) ( ) ( ) α = α = α α-u e f f y x ~ ~ . Let ~ ~ ~ y x z 2-= . Find and sketch the distribution function and the pdf of ~ z . 5. Let ~ x and ~ y be i.i.d. random variables and exponentially distributed with parameter 1. Let ~ ~ ~ y x z + = and ~ ~ x e w = . a) Find the joint density function of ~ z and ~ w and clearly indicate the region involved on the zw-plane. b) Find and sketch the marginal pdf of ~ z . c) Find and sketch the conditional density function ( ) z w f w | ~ for > z ....
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• Spring '09
• Probability theory, Distribution function, probability density function, Middle East Technical University Electrical and Electronics Engineering Department, Department EE230 Homework

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HW5 - = | ~ ~ y y z f z 4 Let ~ x and ~ y be...

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