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Chapter 3: Problem 6

Let \(X_{1}, X_{2}\), and \(X_{3}\) be iid random variables, each with pdf \(f(x)=e^{-x}\), \(0y)=1-P\left(X_{i}>y, i=1,2,3\right)\). (b) Find the distribution of \(Y=\operatorname{maximum}\left(X_{1}, X_{2}, X_{3}\right)\).

Short Answer

Expert verified
The distribution of the minimum of \(X_1, X_2, X_3\) (denoted by \(Y\)) is \(3e^{-y}(1-e^{-y})^2\) and the distribution of the maximum of \(X_1, X_2, X_3\) (also denoted by \(Y\)) is \(3e^{-3y}\).

Step by step solution

01

- Finding Cumulative Distribution Function (CDF)

In the given problem, \(X_1, X_2, X_3\) are iid with pdf \(f(x)=e^{-x}, 0<x<\infty\). The CDF, \(F(x)\) of a random variable \(X_i\) is obtained by integrating the pdf from \(-\infty\) to \(x\). For \(X_i, F(x) = \int_{-\infty}^{x} e^{-t} dt = -e^{-x}\).
02

- Finding the Distribution of the Minimum

The probability that the minimum of \(X_i\) is less or equal to \(y\) is \(1 - P(X_i>y)\). Since all \(X_i\)s are iid, the events \(X_i>y\) are independent. Thus, \(P(X_i>y)\) is same for all \(i=1,2,3\), and equals \(1 - F(y)\). Therefore, \(P(Y \leq y) = 1 - P(X_i > y)^3 = 1 - [1 - (-e^{-y})]^3\). Differentiating \(P(Y \leq y)\) with respect to \(y\), gives the pdf of \(Y\), which is \(3e^{-y}(1-e^{-y})^2\).
03

- Finding the Distribution of the Maximum

The probability that the maximum of \(X_i\) is less or equal to \(y\) is same as the probability that all Xs are less or equal to \(y\), which is \(F(y)\). Since all \(X_i\)s are iid, \(P(Y \leq y) = [F(y)]^3 = (-e^{-y})^3\). Differentiating \(P(Y \leq y)\) with respect to \(y\), gives the pdf of \(Y\), which is \(3(-e^{-y})^2.e^{-y} = 3e^{-3y}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cumulative Distribution Function
Imagine you have a random variable, and you're curious about the probability that this variable will take on a value less than or equal to a certain number. That's where the cumulative distribution function (CDF) comes in. It's a fundamental concept in statistics that shows the probability that a random variable is less than or equal to a specific value. To construct a CDF for a continuous random variable, you integrate its probability density function (PDF) from negative infinity up to the variable's value.

For instance, in our exercise, the CDF for each independent and identically distributed (iid) random variable, given the PDF \(f(x)=e^{-x}\) for \(0
Probability Density Function
Probability density function (PDF) is a statistical expression that defines a probability distribution for a continuous random variable. The PDF provides the probabilities of occurrence of different possible values of the random variable. It's important to note that a PDF itself is not a probability, since for a continuous random variable, the probability at any given exact value is zero. Instead, the probability of the random variable falling within a particular range is given by the integral of the PDF over that range.

In the exercise above, the random variables have the PDF \(f(x)=e^{-x}\), for \(0
Independent and Identically Distributed Random Variables
The concept of independent and identically distributed (iid) random variables is crucial in probability and statistics. These are variables that share the same probability distribution and are mutually independent. This means that the occurrence of one doesn't affect the likelihood of another, and they all behave 'identically' in terms of their statistical properties.

In practice, this assumption simplifies the analysis because you can apply the same rules to each variable. For instance, in the homework problem we're looking at, you're dealing with three iid variables, which means you can calculate the distribution of the minimum or maximum by using the properties of one and extending it across all three, considering their independence. Independence is exploited when we calculate the probability of the minimum or the maximum of these variables by raising their cumulative distribution functions to the power corresponding to the number of variables, due to the multiplicative rule of independent probabilities.

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