Question

11. Suppose that X and Y have joint density function $f(x, y) = \begin{cases} c(1 - x) & , \ 0 < x < y < 1, \\ 0 & , \text{otherwise.} \end{cases}$ (a) Find c. (b) Are X and Y independent? (c) Find $P(X + Y \le .5)$. (d) Find $E(X^3 Y^2)$.

          11. Suppose that X and Y have joint density function
$f(x, y) = \begin{cases} c(1 - x) & , \ 0 < x < y < 1, \\ 0 & , \text{otherwise.} \end{cases}$
(a) Find c.
(b) Are X and Y independent?
(c) Find $P(X + Y \le .5)$.
(d) Find $E(X^3 Y^2)$.

11. Suppose that X and Y have joint density function
f(x, y) = c(1 - x) , 0 < x < y < 1,
0 , otherwise.
(a) Find c.
(b) Are X and Y independent?
(c) Find P(X + Y ≤ .5).
(d) Find E(X^3 Y^2).

Added by April S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Amany Waheeb

02/04/2022

Step 1

∫∫c(1-x)dxdy = 1 We can integrate with respect to x first, since the limits of integration for x depend on y. ∫[0,1]∫[x,1]c(1-x)dydx = 1 Integrating with respect to y, we get: ∫[0,1]c(1-x)(1-x)dx = 1 Simplifying the integral: c∫[0,1](1-x)^2dx = 1 Using the Show more…

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Suppose that X and Y have joint density function: c(1-x)(0<x<y<1), 0 (otherwise). a) Find c. b) Are X and Y independent? c) Find P(X+Y<0.5). d) Find E(X^3Y^2).