The random variables X and Y have joint density function f(x,y)=12xy(1-x) 0&lt;x&lt;1,0&lt;y&lt;1 and equal to 0 otherwise. Are X and Y independent?

It is important to note that the joint PDF of (X,Y) can be written asf(x,y)=12xy(1-x)=6x(1-x)*2y&nbsp;for&nbsp;x∈(0,1)&nbsp;&nbsp;and&nbsp;&nbsp;y∈(0,1), otherwise it is equal to zero. Considering that,∫016x(1−x)&nbsp;dx&nbsp;=1&nbsp;&nbsp;and&nbsp;&nbsp;∫012y&nbsp;dy&nbsp;=1, these two function are in fact marginal functions of X and Y. Hence we have that&nbsp;fX(x)=6x(1−x)&nbsp;fY(y)=2y&nbsp;and since the joint PDF factorizes, these random variables are independent.

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crapthach24

Answered question

2021-11-10

The random variables X and Y have joint density function
f(x,y)=12xy(1-x) 0<x<1,0<y<1
and equal to 0 otherwise.
Are X and Y independent?

Answer & Explanation

Lauren Fuller

Beginner2021-11-11Added 14 answers

It is important to note that the joint PDF of (X,Y) can be written as
$f (x, y) = 12 x y (1 - x) = 6 x (1 - x) * 2 y$
for $x \in (0, 1) and y \in (0, 1)$ , otherwise it is equal to zero. Considering that, $\int_{0}^{1} 6 x (1 - x) d x = 1 and \int_{0}^{1} 2 y d y = 1$ , these two function are in fact marginal functions of X and Y. Hence we have that
$f_{X} (x) = 6 x (1 - x)$
$f_{Y} (y) = 2 y$
and since the joint PDF factorizes, these random variables are independent.

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