**Analysis of Exponentially Distributed Random Variables**Consider a set of independent exponential random variables $X_1, \ldots, X_n$ each with the same rate parameter $\lambda$.**Objective:**Determine the distribution of the minimum of these variables, specifically:a) Identify the distribution of the smallest order statistic $Y_1$.b) Provide the probability density function (pdf) of this distribution and describe the distribution along with its parameters.---In exponential distributions, the minimum of several independent and identically distributed exponential random variables is also an exponential random variable. Thus:- $Y_1$, the minimum, follows an exponential distribution with a new rate parameter $n\lambda$.- The pdf of $Y_1$ is given by:\[ f_{Y_1}(y) = n\lambda e^{-n\lambda y}, \quad y \ge 0 \]This scenario highlights a unique property of exponential distributions pertaining to minimum values, simplifying statistical modeling in numerous real-world applications.

Answered: Let X1, .., Xn all be independent…

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**Analysis of Exponentially Distributed Random Variables**

Consider a set of independent exponential random variables $X_1, \ldots, X_n$ each with the same rate parameter $\lambda$.

**Objective:**
Determine the distribution of the minimum of these variables, specifically:

a) Identify the distribution of the smallest order statistic $Y_1$.

b) Provide the probability density function (pdf) of this distribution and describe the distribution along with its parameters.

---

In exponential distributions, the minimum of several independent and identically distributed exponential random variables is also an exponential random variable. Thus:

- $Y_1$, the minimum, follows an exponential distribution with a new rate parameter $n\lambda$.
- The pdf of $Y_1$ is given by:

\[ f_{Y_1}(y) = n\lambda e^{-n\lambda y}, \quad y \ge 0 \]

This scenario highlights a unique property of exponential distributions pertaining to minimum values, simplifying statistical modeling in numerous real-world applications.

Expert Solution

Step 1

From the given information, the random variables $X_{1}, X_{2}, . . . ., X_{n}$ has an exponentially distributed with parameter $λ$ .

The distribution function for X_i is $F (X_{i}) = 1 - e^{- λ x_{i}}$

The density function for X_i is $f (X_{i}) = λ e^{- λ x_{i}}$

The density function for the smallest order statistic Y1 is,

$g (y 1) = n {[1 - F (X_{i})]}^{n - 1} f (x) = n {[1 - 1 + e^{- λ x}]}^{n - 1} * λ e^{- λ x} = n λ {[e^{- λ x}]}^{n - 1 + 1} = n λ [e^{- n λ x}]$

Hence, the distribution of the smallest order statistic Y1 is Exponential distribution with parameter $n λ$ .

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