FizeauV

Answered question

2021-10-23

Suppose that X and Y are continuous random variables with joint pdf and zero otherwise.
Find P(X>3)

Answer & Explanation

Daphne Broadhurst

Skilled2021-10-24Added 109 answers

Step 1
Introduction:
The joint density function of two random variables X and Y is given below:

The marginal density function of X is,
$f\left(x\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-\left(x+y\right)}dy$
$={e}^{-x}{\int }_{0}^{\mathrm{\infty }}{e}^{-y}dy$
$=-{e}^{-x}{\left[-{e}^{-y}\right]}_{0}^{\mathrm{\infty }}$
$=-{e}^{-x}\left[{e}^{-\mathrm{\infty }}-{e}^{-0}\right]$
$=-{e}^{-x}\left[0,1\right]$
$={e}^{-x}$
Step 2
The probability of P(X > 3) is obtained as 0.0498 from the calculation given below:
$P\left(X>3\right)={\int }_{3}^{\mathrm{\infty }}{e}^{-x}dx$
$={\left[-{e}^{-x}\right]}_{3}^{\mathrm{\infty }}$
$=-\left[{e}^{-\mathrm{\infty }}-{e}^{-3}\right]$
$=-\left[0-{e}^{-3}\right]$
$={e}^{-3}$
=0.0498
Thus, the probability of P(X > 3) is 0.0498.

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