{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

examples-joint-pdfs

# examples-joint-pdfs - Z = X Y Example 5 X and Y are jointly...

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}