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Asked Dec 3, 2019
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6.8. The joint probability density function of X and Y is given by
f(x, y) = c(y² – a?)e
-y
- y <a < y, 0 < y < ∞
a. Find c.
b. Find the marginal densities of X and Y.
c. Find E[X].
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6.8. The joint probability density function of X and Y is given by f(x, y) = c(y² – a?)e -y - y <a < y, 0 < y < ∞ a. Find c. b. Find the marginal densities of X and Y. c. Find E[X].

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Expert Answer

Step 1

2.

The joint probability density function of X and Y is given by,

 

 

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j((x² -x²) e³ , -ysx<y, 0<y<0 [0, f (x,y) ={ otherwise

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Step 2

a).

The value of c is calculated below:

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f(x,y cydx=1 00 ر(آأء-x)c*dbdx = 1 C C 2y =1 edy (a) 24T(4)-1 2cГ (4) Γ (α) - (α -1)! 12c-4c 1 8c =1

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Step 3

b).

The marginal density of X can be...

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-jл 5(х) — [/(х.уo ={(x² -x')•*dy =c[ve*ày-c}xe*ày -сг(3) -сх (1) 2 x 2-х2 :-y<x<y

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