HW - X 1 X 1 X 2(Name it 8 Let X 1 and X 2 have the joint...

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APPM 4/5520 Problem Set One (Due Wednesday, August 31st) 1. Let X be a random variable with the binomial distribution with parameters n and p , (ie: X bin ( n, p )). Find the distribution of Y = n - X . (Name it!) 2. Let X unif (0 , 1). Find the distribution of Y = - ln X . (Name it!) 3. Let X exp(rate = λ ). Find the distribution of Y = e - X . (Name it!) 4. Let X unif (0 , 1). Find the distribution of Y = tan X . (Name it!) 5. Compute the mean of the Γ( α, β ) distribution by “integrating without integrating”. 6. Let X be a continuous random variable with pdf f and cdf F . Let U unif (0 , 1). Show that Y = F - 1 ( U ) has the same distribution as X . 7. Suppose that X 1 , X 2 iid exp ( rate =
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Unformatted text preview: X 1 X 1 + X 2 . (Name it!) 8. Let X 1 and X 2 have the joint pdf f X 1 ,X 2 ( x 1 , x 2 ) = 2 e-x 1-x 2 I (0 ,x 2 ) ( x 1 ) I (0 , ∞ ) ( x 2 ) . (a) Find the marginal pdfs for X 1 and X 2 . (b) Suppose that Y 1 = 2 X 1 and Y 2 = X 2-X 1 . Show that Y 1 and Y 2 are independent. 9. Required for 5520 students only: Suppose that X is a continuous random variable with pdf f ( x ). Let Y = X 2 . (Note that this is not a one-to-one, invertible transformation.) Find an expression for the pdf of Y in terms of the pdf of X ....
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  • Fall '11
  • Manuel
  • Probability distribution, Probability theory, probability density function, Cumulative distribution function, Discrete probability distribution, iid X1 X1

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