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ALI
4. Probability and Statistics - Simple Probability
Let
X have the probability density function given by:
f_(X)(x)=0.5**e^(-|x|) where
-oo < x < oo. What is the probability that
|x| falls between 2 and 4 ? Round your answer to three decimal places.
Pick ONE option
0.117
0.018
0.135
0.068

ALI $\newline$ $4$ . Probability and Statistics - Simple Probability $\newline$ Let $\mathrm{X}$ have the probability density function given by: $\mathrm{f}_{\mathrm{X}}(\mathrm{x})=0.5 * \mathrm{e}^{-|\mathrm{x}|}$ where $-\infty<\mathrm{x}<\infty$ . What is the probability that $|\mathrm{x}|$ falls between $2$ and $4$ ? Round your answer to three decimal places. $\newline$ Pick ONE option $\newline$ $0$ . $117$ $\newline$ $0$ . $018$ $\newline$ $0$ . $135$ $\newline$ $0$ . $068$

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Identify independent events

Full solution

Q. ALI

\newline

4

. Probability and Statistics - Simple Probability

\newline

Let

\mathrm{X}

have the probability density function given by:

\mathrm{f}_{\mathrm{X}}(\mathrm{x})=0.5 * \mathrm{e}^{-|\mathrm{x}|}

where

-\infty<\mathrm{x}<\infty

. What is the probability that

|\mathrm{x}|

falls between

2

and

4

? Round your answer to three decimal places.

\newline

Pick ONE option

\newline

0

.

117

\newline

0

.

018

\newline

0

.

135

\newline

0

.

068

Understand the problem: Understand the problem and the probability density function (pdf) given. $\newline$ The pdf of $X$ is given by $f_X(x) = 0.5 \cdot e^{-|x|}$ , where $-\infty < x < \infty$ . We need to find the probability that $|x|$ falls between $2$ and $4$ . This means we are looking for $P(2 < |x| < 4)$ .
Set up the integral: Set up the integral to calculate the probability. $\newline$ Since the pdf is symmetric around zero, we can calculate the probability for the positive side and then double it. The integral will be from $2$ to $4$ of the pdf. $\newline$ $P(2 < |x| < 4) = 2 \times \int_{2}^{4} f_X(x) \, dx$
Calculate the integral: Calculate the integral. $\newline$ $\int_{2}^{4} f_X(x) \, dx = \int_{2}^{4} 0.5 \cdot e^{-|x|} \, dx$ $\newline$ Since we are integrating from $2$ to $4$ , $|x|$ is just $x$ in this range. $\newline$ $\int_{2}^{4} 0.5 \cdot e^{-x} \, dx = 0.5 \cdot \int_{2}^{4} e^{-x} \, dx$
Solve the integral: Solve the integral. $\newline$ $0.5 \times \int_{2}^{4} e^{-x} dx = 0.5 \times [-e^{-x}]_{2}^{4}$ $\newline$ $= 0.5 \times [-e^{-4} + e^{-2}]$ $\newline$ $= 0.5 \times [e^{-2} - e^{-4}]$
Calculate numerical value: Calculate the numerical value of the integral. $\newline$ $0.5 \times [e^{-2} - e^{-4}] = 0.5 \times [0.1353352832 - 0.0183156388]$ $\newline$ $= 0.5 \times 0.1170196444$ $\newline$ $= 0.0585098222$
Double the result: Double the result since the pdf is symmetric around zero. $\newline$ $P(2 < |x| < 4) = 2 \times 0.0585098222$ $\newline$ $= 0.1170196444$
Round the answer: Round the answer to three decimal places as instructed. $\newline$ $P(2 < |x| < 4) \approx 0.117$

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