Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as f_{X}(t)=f_{Y}(t)=\begin{cases}

Kaycee Roche

Kaycee Roche

Answered question

2021-06-06

Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
fX(t)=fY(t)={2t2, t>20, otherwise
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.

Answer & Explanation

l1koV

l1koV

Skilled2021-06-07Added 100 answers

Step 1
Here, X and Y are independent, continuous random variables.
So, it can be written as: f(x,y)=f(x)f(y)
It is given that,
fX(x)={2x2, x>20, otherwise
fY(y)={2y2, y>20, otherwise
Step 2
(a) The joint probability density function can be obtained as:
fXY(x,y)=fX(x)fY(y)
=2x2×2y2
=4(xY)2
Thus,
fXY(x,y)={4(xy)2, x>2,y>20, otherwise
Step 3
(b)The probability density function of W=XY is same as probability density function of f(x,y).
fW(w)=fXY(x,y)={4(xy)2, x>2,y>20, otherwise

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