If the random variable x is exponentially distributed with parameter $\lambda$ = 1.5,then P(2 ? x ? 4),up to 4 decimal places,is:
A) 0.6667.
B) 0.0473.
C) 0.5000.
D) 0.2500.

Question

If the random variable x is exponentially distributed with parameter $\lambda$ = 1.5,then P(2 	? x ? 4),up to 4 decimal places,is:
A) 0.6667.
B) 0.0473.
C) 0.5000.
D) 0.2500.

Quizplus · Accepted Answer

The probability density function (PDF) of an exponential distribution with parameter $\lambda$ is given by:

$f(x) = {\begin{cases} \lambda e^{-\lambda x} & x \geq 0 \ 0 & x < 0 \end{cases}}$

To find P(2 ≤ x ≤ 4), we need to integrate the PDF from 2 to 4:

$P(2 \leq x \leq 4) = \int_{2}^{4} \lambda e^{-\lambda x} dx$

Using integration by substitution, with u = $\lambda x$ and du = $\lambda dx$, we get:

$P(2 \leq x \leq 4) = \frac{1}{2} \int_{3}^{6} e^{-u} du$

Evaluating the integral, we get:

$P(2 \leq x \leq 4) = \frac{1}{2} (e^{-3} - e^{-6})$

Using a calculator, we get:

$P(2 \leq x \leq 4) \approx 0.0473$

Therefore, the best choice is B.

If the Random Variable X Is Exponentially Distributed with Parameter λ\lambdaλ

If the Random Variable X Is Exponentially Distributed with Parameter $\lambda$