Question

8. Let X and Y be continuous random variables jointly distributed with the joint probability density function f(x, y) = { cx^2y, if 0 <= x <= 1, 0 <= y <= 1 0, otherwise, (a) Find c. (b) Find the marginal probability density functions of X and Y. (c) Find E[X] and E[Y]. (d) Find Var(X) and Var(Y). (e) Find Cov(X, Y). (f) Find Corr(X, Y).

          8. Let X and Y be continuous random variables jointly distributed with the joint
probability density function
f(x, y) = { cx^2y, if 0 <= x <= 1, 0 <= y <= 1
0, otherwise,
(a) Find c.
(b) Find the marginal probability density functions of X and Y.
(c) Find E[X] and E[Y].
(d) Find Var(X) and Var(Y).
(e) Find Cov(X, Y).
(f) Find Corr(X, Y).

8. Let X and Y be continuous random variables jointly distributed with the joint
probability density function
f(x, y) = cx^2y, if 0 <= x <= 1, 0 <= y <= 1
0, otherwise,
(a) Find c.
(b) Find the marginal probability density functions of X and Y.
(c) Find E[X] and E[Y].
(d) Find Var(X) and Var(Y).
(e) Find Cov(X, Y).
(f) Find Corr(X, Y).

Added by Cheyenne M.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

09/03/2023

Step 1

$$\int_0^1 \int_0^1 c \, dy \, dx = 1$$ $$c \int_0^1 \int_0^1 dy \, dx = 1$$ $$c \int_0^1 (1 - 0) \, dx = 1$$ $$c \int_0^1 dx = 1$$ $$c (1 - 0) = 1$$ $$c = 1$$ So, the joint probability density function is $f(x,y) = 1$ for $0 < x < 1$ and $0 < y < 1$. (b) Show more…

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Let X and Y be continuous random variables jointly distributed with the joint probability density function f(x,y) = { cx^2y, if 0 <= x <= 1, 0 <= y <= 1; 0, otherwise. (a) Find c. (b) Find the marginal probability density functions of X and Y. (c) Find E[X] and E[Y]. (d) Find Var(X) and Var(Y). (e) Find Cov(X,Y). (f) Find Corr(X,Y).