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 (A) 0.117 (B) 0.018 (C) 0.135 (D) 0.068

Integrate from 2 to 4: To find the probability that |x| falls between 2 and 4, we need to integrate the probability density function f_(X)(x) from -4 to -2 and from 2 to 4, because |x| between 2 and 4 means x is between -4 and -2 or between 2 and 4.Calculate integral from 2 to 4: First, we calculate the integral from 2 to 4. The integral of f_(X)(x) from 2 to 4 is: ∫ from 2 to 4 of 0.5 * e^(-|x|) dx Since x is positive in this interval, |x| = x, so the integral becomes: ∫ from 2 to 4 of 0.5 * e^(-x) dxCalculate integral value: To solve the integral, we use the antiderivative of e^(-x), which is -e^(-x). Therefore, the integral becomes: -0.5 * e^(-x) evaluated from 2 to 4 = -0.5 * (e^(-4) - e^(-2))Calculate integral from -4 to -2: Now we calculate the value of the integral: -0.5 * (e^(-4) - e^(-2)) = -0.5 * (0.0183 - 0.1353) (using a calculator for e^(-4) and e^(-2)) = -0.5 * (-0.117) = 0.0585Calculate integral value: Next, we calculate the integral from -4 to -2. The integral of f_(X)(x) from -4 to -2 is: ∫ from -4 to -2 of 0.5 * e^(-|x|) dx Since x is negative in this interval, |x| = -x, so the integral becomes: ∫ from -4 to -2 of 0.5 * e^(-(-x)) dx = ∫ from -4 to -2 of 0.5 * e^(x) dxAdd probabilities: To solve this integral, we use the antiderivative of e^(x), which is e^(x). Therefore, the integral becomes: 0.5 * e^(x) evaluated from -4 to -2 = 0.5 * (e^(-2) - e^(-4))Now we calculate the value of the integral: 0.5 * (e^(-2) - e^(-4)) = 0.5 * (0.1353 - 0.0183) (using a calculator for e^(-2) and e^(-4)) = 0.5 * 0.117 = 0.0585Finally, we add the probabilities from the two intervals to find the total probability that |x| falls between 2 and 4: 0.0585 (from 2 to 4) + 0.0585 (from -4 to -2) = 0.117

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Let $X$ have the probability density function given by: $\newline$ $f_{X}(x)=0.5e^{-|x|}$ where -\infty < x < \infty. $\newline$ What is the probability that $|x|$ falls between $2$ and $4$ ? $\newline$ Round your answer to three decimal places. $\newline$ Pick ONE option $\newline$ (A) $0.117$ $\newline$ (B) $0.018$ $\newline$ (C) $0.135$ $\newline$ (D) $0.068$

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Precalculus

Identify independent events

Full solution

Q. Let

X

have the probability density function given by:

\newline

f_{X}(x)=0.5e^{-|x|}

where

-\infty < x < \infty

\newline

What is the probability that

|x|

falls between

2

and

4

\newline

Round your answer to three decimal places.

\newline

Pick ONE option

\newline

(A)

0.117

\newline

(B)

0.018

\newline

(C)

0.135

\newline

(D)

0.068

Integrate from $2$ to $4$ : To find the probability that $|x| ) falls between \$2$ and $4$ , we need to integrate the probability density function $f_{X}(x)$ from $-4$ to $-2$ and from $2$ to $4$ , because $|x|$ between $2$ and $4$ means $4$ $0$ is between $-4$ and $-2$ or between $2$ and $4$ .
Calculate integral from $2$ to $4$ : First, we calculate the integral from $2$ to $4$ . The integral of $f_{X}(x)$ from $2$ to $4$ is: $\newline$ $\int_{2}^{4} 0.5 \cdot e^{-|x|} \, dx$ $\newline$ Since $x$ is positive in this interval, $|x| = x$ , so the integral becomes: $\newline$ $\int_{2}^{4} 0.5 \cdot e^{-x} \, dx$
Calculate integral value: To solve the integral, we use the antiderivative of $e^{-x}$ , which is $-e^{-x}$ . Therefore, the integral becomes: $\newline$ $-0.5 \times e^{-x}$ evaluated from $2$ to $4$ $\newline$ $= -0.5 \times (e^{-4} - e^{-2})$
Calculate integral from $-4$ to $-2$ : Now we calculate the value of the integral: $\newline$ $-0.5 \times (e^{-4} - e^{-2}) = -0.5 \times (0.0183 - 0.1353)$ (using a calculator for $e^{-4}$ and $e^{-2}$ ) $\newline$ $= -0.5 \times (-0.117)$ $\newline$ $= 0.0585$
Calculate integral value: Next, we calculate the integral from $-4$ to $-2$ . The integral of $f_{X}(x)$ from $-4$ to $-2$ is: $\newline$ $\int_{-4}^{-2} 0.5 \cdot e^{(-|x|)} \, dx$ $\newline$ Since $x$ is negative in this interval, $|x| = -x$ , so the integral becomes: $\newline$ $\int_{-4}^{-2} 0.5 \cdot e^{(-(-x))} \, dx$ $\newline$ $= \int_{-4}^{-2} 0.5 \cdot e^{x} \, dx$
Add probabilities: To solve this integral, we use the antiderivative of $e^{x}$ , which is $e^{x}$ . Therefore, the integral becomes: $\newline$ $0.5 \times e^{x}$ evaluated from $-4$ to $-2$ $\newline$ = $0.5 \times (e^{-2} - e^{-4})$
Add probabilities: To solve this integral, we use the antiderivative of $e^{x}$ , which is $e^{x}$ . Therefore, the integral becomes: $\newline$ $0.5 \times e^{x}$ evaluated from $-4$ to $-2$ $\newline$ = $0.5 \times (e^{-2} - e^{-4})$ Now we calculate the value of the integral: $\newline$ $0.5 \times (e^{-2} - e^{-4}) = 0.5 \times (0.1353 - 0.0183)$ (using a calculator for $e^{-2}$ and $e^{-4}$ ) $\newline$ = $0.5 \times 0.117$ $\newline$ = $e^{x}$ $0$
Add probabilities: To solve this integral, we use the antiderivative of $e^{x}$ , which is $e^{x}$ . Therefore, the integral becomes: $\newline$ $0.5 \times e^{x}$ evaluated from $-4$ to $-2$ $\newline$ = $0.5 \times (e^{-2} - e^{-4})$ Now we calculate the value of the integral: $\newline$ $0.5 \times (e^{-2} - e^{-4}) = 0.5 \times (0.1353 - 0.0183)$ (using a calculator for $e^{-2}$ and $e^{-4}$ ) $\newline$ = $0.5 \times 0.117$ $\newline$ = $e^{x}$ $0$ Finally, we add the probabilities from the two intervals to find the total probability that $e^{x}$ $1$ falls between $e^{x}$ $2$ and $e^{x}$ $3$ : $\newline$ $e^{x}$ $0$ (from $e^{x}$ $2$ to $e^{x}$ $3$ ) + $e^{x}$ $0$ (from $-4$ to $-2$ ) = $0.5 \times e^{x}$ $0$