Let X and Y be two continuous random variables with the joint density function f(x,y) = {x – 2y, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 0, elsewhere (a) Are the random variables X and Y independent? Justify your answer. (b) Compute the numerical value of P(Y ≥ 1, X ≤ ¹).

Let X and Y be two continuous random variables with the joint density function f(x,y) = {x – 2y, 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 0, elsewhere (a) Are the random variables X and Y independent? Justify your answer. (b) Compute the numerical value of P(Y ≥ 1, X ≤ ¹).

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

100%

Let X and Y be two continuous random variables with the joint density function
f(x,y) = {x
2y, 0≤ x ≤ 1 and 0 ≤ y ≤ 1
0, elsewhere
(a) Are the random variables X and Y independent? Justify your answer.
(b) Compute the numerical value of P(Y ≥ 1, X ≤ ¹1).

Expert Solution

Step 1:- (a)

X and Y are independent if ,

f(x,y)= f(x)*f(y)

Given that, joint pdf of x and y is,

f(x,y)= (x-2y) ; for 0<= x<=1 and 0<=y<=1

Marginal pdf of X;

f(x)= ∫_y f(x,y)dy

= ∫₀¹ (x-2y)dy

= (xy-y²)₀¹

= (x-1)

f(x)= (x-1)

Step by step

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