examples-joint-pdfs

examples-joint-pdfs - Z = X + Y . Example 5: X and Y are...

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AMS 311 Joe Mitchell More Examples: Joint Densities Example 1: X and Y are jointly continuous with joint pdf f ( x,y ) = ± cx 2 + xy 3 if 0 x 1, 0 y 2 0 , otherwise. (a). Find c . (b). Find P ( X + Y 1). (c). Find marginal pdf’s of X and of Y . (d). Are X and Y independent (justify!). (e). Find E ( e X cos Y ). (f). Find cov ( X,Y ). Example 2: X and Y are jointly continuous with joint pdf f ( x,y ) = ± cxy if 0 x , 0 y , x + y 1 0 , otherwise. (a). Find c . (b). Find P ( X + Y 1). (c). Find marginal pdf’s of X and of Y . (d). Are X and Y independent (justify!). Example 3: X and Y are jointly continuous with joint pdf f ( x,y ) = ± cxy if 0 x 1, 0 y 1 0 , otherwise. (a). Find c . (b). Find P ( | Y - 2 X | ≤ 0 . 1). (c). Find marginal pdf’s of X and of Y . (d). Are X and Y independent (justify!). Example 4: X and Y are independent continuous random variables, each with pdf g ( w ) = ± 2 w if 0 w 1 0 , otherwise. (a). Find P ( X + Y 1). (b). Find the cdf and pdf of
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Unformatted text preview: Z = X + Y . Example 5: X and Y are jointly continuous with joint pdf f ( x,y ) = ± e-( x + y ) if 0 ≤ x , 0 ≤ y , otherwise. Let Z = X/Y . Find the pdf of Z . Example 6: X and Y are independent, each with an exponential( λ ) distribution. Find the density of Z = X + Y and of W = Y-X 2 . Example 7: X and Y are jointly continuous with ( X,Y ) uniformly distributed over the union of the two squares { ( x,y ) : 0 ≤ x ≤ 1 , 1 ≤ y ≤ 1 } and { ( x,y ) : 0 ≤ x ≤ 1 , 3 ≤ y ≤ 4 } . (a). Find E ( Y ). (b). Find the marginal densities of X and Y . (c). Are X and Y independent? (d). Find the pdf of Z = X + Y ....
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This note was uploaded on 06/06/2010 for the course EC 11 taught by Professor All during the Spring '10 term at UCLA.

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