a6 - Y 2 =-sin θX 1 + cos θX 2 have the same joint...

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Math 4710: Homework 6 Due Friday, October 17, 2008 The primary references for this week’s assignment are Prof. Durrett’s Essen- tials of Probability sections 3.6 and 3.7 as well as Prof. Ross’ First Course in Probability section 6.7 in the 7th edition. Both of these books are on reserve. Joint Distributions Please hand in: problems 29, 30, 33, 36, and 37 in Section 5.6 of EPFA . Extra Problems 1. Suppose X 1 and X 2 have a joint density given by f ( x 1 , x 2 ) = 1 when both 0 < x 1 < 1 and 0 < x 2 < 1, but 0 otherwise. Find the joint density of Y 1 = X 1 X 2 and Y 2 = X 1 X 2 . 2. Suppose that X 1 and X 2 have joint density f ( x 1 , x 2 ) = λ m + n +2 x m 1 x n 2 m ! n ! e - λ ( x 1 + x 2 ) . Find the joint density of Y 1 = X 1 + X 2 and Y 2 = X 1 X 1 + X 2 . 3. Suppose that X 1 and X 2 have joint density f ( x 1 , x 2 ) = g ± q x 2 1 + x 2 2 ² . Show that Y 1 = cos θX 1 + sin θX 2 and
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Unformatted text preview: Y 2 =-sin θX 1 + cos θX 2 have the same joint distribution. In words, this distribution is invariant under rotation. 4. Suppose X 1 , . . . , X n are independent and have density function f ( z ) . Let Y = min ( X 1 , . . . , X n ) and Z = max ( X 1 , . . . , X n ) . Compute P ( Y ≥ y, Z ≤ z ) and differentiate to find the joint density of Y and Z . 5. Suppose X and Y are independent and have density function f ( x ) = 2 x for 0 < x < 1. Find the density function of X + Y. 6. Suppose X and Y are independent and have exponential distributions with λ < μ. Find the density function of X + Y....
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This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell.

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