Question

3. Suppose X and Y are continuous random variables with joint PDF given by f_{X,Y}(x,y) = { 6xy 0 ? x ? 1, 0 ? y ? ?x 0 otherwise a) Find f_X(x) b) Find f_Y(y) c) Are X and Y independent? d) Find the conditional PDF of X given Y=y: f_{X|Y}(x|y) e) Find E(X|Y=y) for 0 ? y ? 1 f) Find Var(X|Y=y) for 0 ? y ? 1

          3. Suppose X and Y are continuous random variables with joint PDF given by
f_{X,Y}(x,y) = {
6xy 0 ? x ? 1, 0 ? y ? ?x
0 otherwise
a) Find f_X(x)
b) Find f_Y(y)
c) Are X and Y independent?
d) Find the conditional PDF of X given Y=y: f_{X|Y}(x|y)
e) Find E(X|Y=y) for 0 ? y ? 1
f) Find Var(X|Y=y) for 0 ? y ? 1

3. Suppose X and Y are continuous random variables with joint PDF given by
fX,Y(x,y) =
6xy 0 ? x ? 1, 0 ? y ? ?x
0 otherwise
a) Find fX(x)
b) Find fY(y)
c) Are X and Y independent?
d) Find the conditional PDF of X given Y=y: fX|Y(x|y)
e) Find E(X|Y=y) for 0 ? y ? 1
f) Find Var(X|Y=y) for 0 ? y ? 1

Added by Leah C.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

06/15/2023

Step 1

To find the marginal PDF of X, we need to integrate the joint PDF with respect to Y: $$ f_X(x) = \int_{0}^{\sqrt{x}} 6xr dy $$ Integrating with respect to Y, we get: $$ f_X(x) = 6xr\left[\frac{y^2}{2}\right]_0^{\sqrt{x}} = 6xr\left(\frac{x}{2}\right) = Show more…

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Suppose X and Y are continuous random variables with joint PDF given by 6xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ ∑x. Find fX(x) and fY(y). Are X and Y independent? Find the conditional PDF of X given Y=y: fX|Y(x|y). Find E(X|Y=y) for 0 ≤ y ≤ 1. Find Var(X|Y=y) for 0 ≤ y ≤ 1.