Question: "Suppose that X and Y are continuous random..."

nagasenaz

Answered question

2021-05-23

Suppose that X and Y are continuous random variables with joint pdf $f (x, y) = e^{- (x + y)} 0$ and zero otherwise.
Find $P (X > 3)$

Answer & Explanation

Arnold Odonnell

Skilled2021-05-24Added 109 answers

Step 1
Introduction:
The joint density function of two random variables X and Y is given below:
$f (x, y) = {\begin{matrix} e^{- (x + y)} . 0 \leq x \leq \infty, 0 \leq y < \infty \\ 0. e l s e w h e r e \end{matrix}}$
The marginal density function of X is,
$f (x) = \int_{0}^{\infty} e^{- (x + y)} d y$
$= e^{- x} \int_{0}^{\infty} e^{- y} d y$
$= - e^{- x} [- e^{- y}]_{0}^{\infty}$
$= - e^{- x} [e^{- \infty} - e^{- 0}]$
$= - e^{- x} [0, 1]$
$= e^{- x}$
Step 2
The probability of $P (X > 3)$ is obtained as 0.0498 from the calculation given below:
$P (X > 3) = \int_{3}^{\infty} e^{- x} d x$
$= [- e^{- x}]_{3}^{\infty}$
$= - [e^{- \infty} - e^{- 3}]$
$= - [0 - e^{- 3}]$
$= e^{- 3}$
=0.0498
Thus, the probability of $P (X > 3)$ is 0.0498.

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