Question

Let X ~ Exp(?), i.e. the random variable X follows the exponential distribution with parameter ? > 0. The probability density function f of the random variable X is defined for all x ? ?: f(x; ?) = { ? exp(-?x) if x ? 0 0 if x < 0 } 1. Find the cumulative distribution function F of the random variable X. 2. Compute Pr(0 ? X ? 5). 3. Compute the expectation of the random variable X. Find the variance & the standard deviation of the random variable X. 4. Find the median of the random variable X.

          Let X ~ Exp(?), i.e. the random variable X follows the exponential distribution with parameter ? > 0. The probability density function f of the random variable X is defined for all x ? ?:
f(x; ?) = {
? exp(-?x) if x ? 0
0 if x < 0
}
1. Find the cumulative distribution function F of the random variable X.
2. Compute Pr(0 ? X ? 5).
3. Compute the expectation of the random variable X. Find the variance & the standard deviation of the random variable X.
4. Find the median of the random variable X.

Let X Exp(?), i.e. the random variable X follows the exponential distribution with parameter ? > 0. The probability density function f of the random variable X is defined for all x ? ?:
f(x; ?) =
? exp(-?x) if x ? 0
0 if x < 0

1. Find the cumulative distribution function F of the random variable X.
2. Compute Pr(0 ? X ? 5).
3. Compute the expectation of the random variable X. Find the variance the standard deviation of the random variable X.
4. Find the median of the random variable X.

Added by Angela V.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

08/03/2023

Step 1

To find the cumulative distribution function (CDF) F(x) of the random variable X, we need to integrate the probability density function (PDF) f(x) from -∞ to x: F(x) = ∫f(t) dt from -∞ to x Since f(t) = 0 for t < 0, we only need to integrate from 0 to x: F(x) = Show more…

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Let X ~ Exp(λ), i.e. the random variable X follows the exponential distribution with parameter λ > 0. The probability density function f of the random variable X is defined for all x ∈ ℝ: f(x; λ) = { λ exp(-λx) if x ≥ 0 0 if x < 0 } 1. Find the cumulative distribution function F of the random variable X. 2. Compute Pr(0 ≤ X ≤ 5). 3. Compute the expectation of the random variable X. Find the variance & the standard deviation of the random variable X. 4. Find the median of the random variable X.