Question

Suppose the set of possible values for (X, Y ) is the rectangle D = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. Let the joint probability density function of (X, Y ) be f(x, y) = 6 5 (x + y 2 ), for (x, y) ∈ D. a) Verify that f(x, y) is a valid probability density function b) Find P(0 ≤ X ≤ 1/4, 0 ≤ Y ≤ 1/4). c) Find the marginal pdf of X and Y. d) Find P(1/ 4 ≤ Y ≤ 3 /4 )  

Answer 1

Given pdf is-

$f(x)=\dfrac{6}{5}(x+y^2)$

The joint distribution table is given by-

(a) As the values of probabilities is positive so $f(x,y)$ is valid probability density function.

(b) $P(0\le X\le \dfrac{1}{4},0\le Y\le \dfrac{1}{4})=0$

(c) Marginal pdf of X and Y is-

P(X)

P(Y)

(d) $P(\dfrac{1}{ 4} ≤ Y ≤ \dfrac{3 }{4} )=\dfrac{12}{5}-\dfrac{6}{5}=\dfrac{6}{5}$