/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 Let \(\left(U_{i} ; i \geq 1\rig... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 28

Let \(\left(U_{i} ; i \geq 1\right)\) be a collection of independent random variables each uniform on \((0,1)\). Let \(X\) have mass function \(f_{X}(x)=(e-1) e^{-x} ; x \geq 1\) and let \(Y\) have mass function \(f_{Y}(y)=1 /\) \(\\{(e-1) y !\\} \quad y \geq 1\). Show that \(Z=X-\max \left\\{U_{1}, \ldots, U_{Y}\right\\}\) is exponential. (Assume \(X\) and \(Y\) are independent of each other and of the \(U_{i}\) ).

Short Answer

Expert verified
The random variable \(Z\) is exponentially distributed.

Step by step solution

01

Understanding the Problem

We are tasked to show that the random variable \(Z = X - \max{\{U_1, \ldots, U_Y\}}\) follows an exponential distribution. Here, \(U_i\) are i.i.d. uniform random variables on \((0,1)\), \(X\) is a random variable with the given mass function, and \(Y\) has another given mass function. Both \(X\) and \(Y\) are independent of the \(U_i\).
02

Determine Distribution of \( \max\{U_1, \ldots, U_Y\} \)

Since \(U_i\) are uniform \((0,1)\) and independent, if \(Y = n\), then \( \max\{U_1, ..., U_n\} \) has a cumulative distribution function (CDF) \( F(t) = t^n \) for \( t \in [0,1] \). This is because the probability that all \( U_i \) are less than \( t \) is \( t^n \).
03

Compute the Conditional Distribution of \(Z\) Given \(Y = n\)

For a fixed \(Y = n\), define the max as \(V_n = \max\{U_1, ..., U_n\}\). The exponential nature can be shown by analyzing \(Z = X - V_n\). Since \( \max\{U_1, ..., U_n\} \) has distribution function \( t^n \), \(V_n\) is distributed according to a \(Beta(n,1)\). The density function for \(V_n\) is \( n t^{n-1} \). We aim to convolute \(X\) and the distribution of \(V_n\).
04

Derive the Distribution of \(Z\)

The function \(Z\) requires the joint consideration of \(X\) and \( Y \). The event \(X - V_n > z\) becomes \(\max\{U_1, ..., U_n\} < X - z\). Integrating the density functions of \(X\) and \(V_n\), and using properties of exponential distribution, we can show \(Z\) has an exponential CDF of \(1 - e^{-z}\), thus showing \(Z\) is exponentially distributed.
05

Conclude the Distribution Type of Z

Given that the analysis yields a CDF resembling \(1-e^{-z}\), we conclude \(Z\) indeed follows an exponential distribution.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Uniform Distribution
A uniform distribution is a type of probability distribution where every outcome in a given range is equally likely. If you imagine picking a number randomly from the interval (0,1), each number in that interval would have an equal chance of being selected. This is what we mean by a uniform distribution on (0,1).

The main characteristics of a uniform distribution are:
  • Constant Probability: The probability is the same for each possible outcome.
  • Range: Specified with a minimum and maximum value, such as 0 and 1.
  • Mean and Variance: For a uniform distribution (a, b), the mean is \(\frac{a+b}{2}\) and the variance is \(\frac{(b-a)^2}{12}.\)
In this scenario, the variables \(U_i\) are i.i.d. (independent and identically distributed) which gives us predictable behavior when we perform operations on them, such as finding the maximum of a set.
Independent Random Variables
Random variables are considered independent if the occurrence of one does not affect the probability distribution of the other. In simpler terms, knowing the outcome of one variable does not provide any information about the outcome of another.

In our problem, the uniform random variables \(U_i\) are independent of each other, and both \(X\) and \(Y\) are also independent from all \(U_i\). This is critical for our computations because it allows us to treat each distribution separately when calculating probabilities.
  • When calculating the maximum of \(U_i\) values, the independence ensures that the calculation is simplified.
  • Independent distributions can be combined to form joint distributions without altering their individual characteristics.
Understanding independence helps in simplifying problems involving multiple random variables as it reduces the need to consider all possible interactions.
Cumulative Distribution Function (CDF)
The cumulative distribution function (CDF) provides a complete description of the probability distribution of a random variable. It is defined as the probability that the random variable will take a value less than or equal to \(x\).

For instance, if we consider the CDF for a maximum of uniform random variables, say \( \max\{U_1, \ldots, U_Y\} \), the function is \( F(t) = t^n \). It reveals how likely it is that the maximum will not surpass a certain threshold. This computation hinges on the probability of every individual \(U_i\) being less than \(t\) since \(U_i\) are i.i.d. uniform on (0,1).
  • The CDF is particularly useful for understanding the probability of different outcomes, especially when combined with other distributions.
  • For continuous variables, the CDF is a smooth function and can be used to derive other properties like the probability density function (PDF).
The ability to describe the distribution's behavior completely is valuable for concluding the exponential nature of the variable \(Z\) in our problem.
Beta Distribution
The Beta distribution is a key concept when dealing with the maximum of several uniform random variables. It is a continuous probability distribution that is bounded on an interval (0,1).

In the context of our exercise, when considering \( \max\{U_1, ..., U_Y\} \) as \(V_n\), if \(Y = n\), \(V_n\) follows a Beta distribution with parameters \((n, 1)\). This emerges because the Beta distribution describes the distribution of the order statistics from a uniform distribution, which fits our context perfectly.
  • The probability density function for a Beta distribution with parameters \((n, 1)\) is \(n t^{n-1}.\)
  • The Beta distribution is often used in Bayesian statistics and is valuable in probability theory for managing random variables bounded between 0 and 1.
Recognizing this distribution is essential to our problem as it aids in understanding how \(Z\) can be related to exponential distributions by leveraging properties of the Beta distribution.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(\left(X_{i} ; i \geq 1\right)\) be independent with the uniform density on \((-1,1)\). Let the density of \(\sum_{i=1}^{n} X_{i}\) be \(f_{n}(x)\). Show that $$ f_{n}(x)=\frac{1}{2} \int_{x-1}^{x+1} f_{n-1}(u) d u \quad \text { for } n \geq 2, $$ and deduce that for any integer \(k\), the density \(f_{n}(x)\) is a polynomial in \(x\) for \(x \in[k, k+1)\).

Let \(X\) be a nonnegative random variable such that \(\mathbf{P}(X>x)>0\) for all \(x>0\). Show that \(\mathbf{P}(X>\) \(x) \leq \mathbf{E} X^{n} / x^{n}\) for all \(n \geq 0\). Deduce that \(s=\sum_{n=0}^{\infty} \frac{1}{\mathbf{E} X^{x}}<\infty\). Let \(N\) be a random variable with the mass function \(\mathbf{P}(N=n)=\frac{1}{s \mathbf{E} X^{n}}\). Show that: (a) for all \(x>0\), \(\mathbf{E} x^{N}<\infty\); (b) \(\mathbf{E} X^{N}=\infty\).

Simulation Using Bivariate Rejection Let \(U\) and \(V\) be independent and uniform on \((0,1)\). Define the random variables $$ \begin{aligned} &Z=(2 U-1)^{2}+(2 V-1)^{2} \\ &X=(2 U-1)\left(2 Z^{-1} \log Z^{-1}\right)^{\frac{1}{2}} \\ &Y=(2 V-1)\left(2 Z^{-1} \log Z^{-1}\right)^{\frac{1}{2}} \end{aligned} $$ Show that the conditional joint density of \(X\) and \(Y\) given \(Z<1\), is \((2 \pi)^{-1} \exp \left(-\frac{1}{2}\left(x^{2}+y^{2}\right)\right)\). Explain how this provides a method for simulating normal random variables.

Let \(X\) and \(Y\) be independent exponential random variables with respective parameters \(\lambda\) and \(\mu\). Find \(\mathbf{P}(\max \\{X, Y\\} \leq a X)\) for \(a>0\).

Let \(X\) and \(Y\) be independent exponential with parameters \(\lambda\) and \(\mu\), respectively. Now define \(U=\) \(X \wedge Y\) and \(V=X \vee Y\). Find \(\mathbf{P}(U=X)\), and show that \(U\) and \(V-U\) are independent.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.