"Let the continuous random variables X and Y..." was answered by our community

Ayaana Buck

Answered question

2021-05-13

Let the continuous random variables X and Y have joint pdf

$f (x, y) = e^{- x - y}, 0$

a) $e^{- y}$

b) $e^{- x}$

c) $\frac{e^{y}}{e^{- x}}$

Answer & Explanation

Daphne Broadhurst

Skilled2021-05-14Added 109 answers

Step 1
The joint pdf of the continuous random variables X and Y is given by :
$f (x, y) = e^{- x - y}, 0$
Here we have to find the value of $f (\frac{y}{x})$ .
As we know that $f (\frac{y}{x}) = \frac{f (x, y)}{f_{X} (x)}$
And $f_{X} (x) = \int_{a}^{b} f (x, y) d y$
Step 2
Let us finding the required value:
$f (\frac{y}{x}) = \frac{f (x, y)}{\int_{a}^{b} f (x, y) d y}$
$= \frac{e^{- x - y}}{\int_{0}^{\infty} e^{- x - y} d y}$
$= \frac{e^{- x - y}}{[- e^{- x - y}]_{0}^{\infty}}$
$= \frac{e^{- x - y}}{[- e^{- \infty} e^{- x}]}$
$= \frac{e^{- x - y}}{e^{- x}}$
$= e^{- x - y + x}$
$= e^{- y}$
Thus, the required value of $f (\frac{y}{x}) = e^{- y}$ .
Hence, option (a) is correct.

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