Homework4 - ST561 Fall 2010 Homework 4 Due Monday Oct 25th...

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ST561 Fall 2010 Homework 4 Due: Monday, Oct. 25th, 2010 1. Choose a point at random from the interval (0 , 1) and call this random variable X 1 . If x 1 is the observed value of X 1 , choose a point at random from the interval (0 , x 1 ) and call this random variable X 2 . (a) Give the joint p.d.f. of ( X 1 , X 2 ). Sketch the region in the plane that is the support of ( X 1 , X 2 ). (b) Give the marginal p.d.f. of X 2 . (c) Give the conditional p.d.f. f ( x 2 /x 1 ) of X 2 given x 1 . (d) Compute P ( X 1 + X 2 1). (e) Find the conditional mean E ( X 1 | X 2 = x 2 ). (f) Find P (0 . 25 X 2 0 . 5 | X 1 0 . 5). 2. Let X 1 and X 2 be independent random variables with
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Unformatted text preview: X 1 being binomial( n = 3 ,p = 1 / 3) and X 2 being binomial( n = 4 ,p = 1 / 2). Compute P ( X 1 = X 2 ). 3. Textbook Page 161, 3.2.12 4. Suppose the distribution of X 2 condition on X 1 = x 1 is N ( x 1 ,x 2 1 ) and that the marginal distribution of X 1 is U (0 , 1). Find the mean and variance of X 2 . 5. Suppose X 1 is distributed as N ( μ,σ 2 ) and conditionally on X 1 = x 1 , X 2 is distributed as N ( x 1 ,σ 2 ). Find the marginal distribution of X 2 . 1...
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  • Fall '08
  • Staff
  • Probability theory, X1, X2 given X1

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