/*! 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 14 Let \(X\) be an exponential rand... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 14

Let \(X\) be an exponential random variable with rate \(\lambda\). (a) Use the definition of conditional expectation to determine \(E[X \mid Xc] P\\{X>c\\} $$

Short Answer

Expert verified
In summary, for an exponential random variable \(X\) with rate \(\lambda\), we find the conditional expectation \(E[X \mid X<c]\) using two methods: (a) Direct calculation using truncated distribution yields: \(E[X \mid X<c] = \frac{1}{\lambda}\). (b) Applying the Law of Total Expectation also gives: \(E[X \mid X<c] = \frac{1}{\lambda}\).

Step by step solution

01

Understanding the Exponential Distribution

The exponential distribution is a statistical distribution used to model the time elapsed between events. It is defined by its rate parameter \(\lambda\), which is the average number of events per unit time. The probability density function (PDF) of an exponential random variable is given by \(f(x)=\lambda e^{-\lambda x}\) for \(x \geq 0\), and E[X] = \(\frac{1}{\lambda}\).
02

Defining Conditional Expectation

Conditional expectation, or the expected value of a random variable given some condition, is described by integrating over the region of interest, the product of the variable and its conditional distribution. It is given as \(E[X|B] = \int_{B} xf(x)dx \) for any event B that has positive probability.
03

Solve for \( E[X \mid X

To calculate \(E[X \mid X<c]\) we will use the truncated distribution which describes the random variable \(X\) given \(X<c\). In mathematical terms, this is \(\int_{0}^{c} x \cdot f(x) dx\), where \(f(x) = \lambda e^{-\lambda x }\). Integrate by parts (let \(u=x\) and \(dv=e^{-\lambda x}\)dx to get \(du=dx\), \(v=-\frac{1}{\lambda}e^{-\lambda x }\)) to get \(-\frac{x}{\lambda} e^{-\lambda x} \Biggr|_0^{c} + \int_{0}^{c} \frac{1}{\lambda} e^{-\lambda x } dx\). Evaluate this to find that \(E[X \mid X<c] = \frac{1}{\lambda}\).
04

Use the Law of Total Expectation to solve for \( E[X \mid X

The Law of Total Expectation allows us to compute \(E[X \mid Xc] P\{X>c\}\). We already know that \(E[X] = \frac{1}{\lambda}\) and \(P\{Xc\} = e^{-\lambda c}\). For \(E[X \mid X>c]\), it can be shown that for any exponential distribution with rate \(\lambda\) if \(X>c\), then \(X-c\) is also exponentially distributed with rate \(\lambda\). Hence, \(E[X \mid X>c] = c + \frac{1}{\lambda}\). By substituting all these values into the initial equation, and solving for \(E[X \mid X<c]\), we also get that \(E[X \mid X<c] = \frac{1}{\lambda}\).

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.

Exponential Random Variable
An exponential random variable is commonly used to describe the time until a certain event occurs, such as the failure of a machine or the arrival of a customer in a queue. It is defined by a key characteristic, namely its continuous nature and the property of memorylessness, meaning the future probability of an event occurring does not depend on how much time has already elapsed.

This distribution is particularly useful in the field of reliability engineering and telecommunications to model lifetimes of objects or the time between phone calls, respectively. The memorylessness property makes the exponential distribution unique among the continuous probability distributions.
Rate Parameter \(\lambda\)
The rate parameter \(\lambda\) in an exponential distribution is a positive real number that represents the average number of events occurring in a given time interval. It defines the frequency with which events are expected to happen.

For example, if \(\lambda=2\), it indicates on average, two events are expected per unit time. As \(\lambda\) increases, the distribution becomes more 'steep' suggesting events occur more frequently. Conversely, a smaller \(\lambda\) indicates a 'flatter' distribution where events occur less frequently. Moreover, the expected value of an exponential random variable is inversely related to \(\lambda\), given by \(E[X]=\frac{1}{\lambda}\).
Probability Density Function (PDF)
The Probability Density Function (PDF) is a function used to specify the probability of a continuous random variable falling within a particular range of values. For the exponential random variable, the PDF is defined for \(x \geq 0\) and is given by \(f(x)=\lambda e^{-\lambda x}\).

The PDF represents the likelihood of different outcomes and is useful for calculating probabilities and expected values for continuous random variables. It is integral to understanding and working with any continuous distribution, such as the exponential distribution we're discussing.
Law of Total Expectation
The Law of Total Expectation is a fundamental theorem in probability theory. It states that the expected value of a random variable can be expressed as the weighted average of expected values of that variable across different, non-overlapping subgroups of a sample space.

In essence, it allows us to break down complex expected value calculations into simpler parts. This is particularly useful in conditional expectation problems, where we may not know the detailed probability structure of the whole space, but we do have information about its partitions.
Truncated Distribution
A truncated distribution arises when a distribution is 'cut off' or 'truncated' at a certain point. For an exponential random variable, we often analyze a truncated distribution to evaluate the expected value or variance within a limited range, such as \(X
This concept is essential when dealing with situations where we know a certain event hasn't occurred within a specified time frame and we want to recalculate probabilities and expectations within this new context. It's like redrawing the normal limits to focus on a specific interval, which provides a more tailored analysis of the probabilities involved.

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

Consider an infinite server queueing system in which customers arrive in accordance with a Poisson process and where the service distribution is exponential with rate \(\mu\). Let \(X(t)\) denote the number of customers in the system at time \(t\). Find (a) \(E[X(t+s) \mid X(s)=n]\) (b) \(\operatorname{Var}[X(t+s) \mid X(s)=n]\) Hint: Divide the customers in the system at time \(t+s\) into two groups, one consisting of "old" customers and the other of "new" customers.

A doctor has scheduled two appointments, one at 1 P.M. and the other at \(1: 30 \mathrm{P}, \mathrm{M}\). The amounts of time that appointments last are independent exponential random variables with mean 30 minutes. Assuming that both patients are on time, find the expected amount of time that the \(1: 30\) appointment spends at the doctor's office.

A flashlight needs two batteries to be operational. Consider such a flashlight along with a set of \(n\) functional batteries-battery 1 , battery \(2, \ldots\), battery \(n\). Initially, battery 1 and 2 are installed. Whenever a battery fails, it is immediately replaced bythe lowest numbered functional battery that has not yet been put in use. Suppose that the lifetimes of the different batteries are independent exponential Trandom variables each having rate \(\mu\). At a random time, call it \(T\), a battery will fail andour stockpile will be empty. At that moment exactly one of the batteries - which we call battery \(X\) -will not yet have failed. (a). What is \(P\\{X=n\\} ?\) (b) What is \(P(X=1) ?\) (c) What is \(P(X=i\\} ?\) (d) Find \(E[T]\). (e) What is the distribution of \(T ?\)

A cable car starts off with \(n\) riders. The times between successive stops of the car are independent exponential random variables with rate \(\lambda\). At each stop one rider gets off. This takes no time, and no additional riders get on. After a rider gets off the car, he or she walks home. Independently of all else, the walk takes an exponential time with rate \(\mu\). (a) What is the distribution of the time at which the last rider departs the car? (b) Suppose the last rider departs the car at time \(t\). What is the probability that all the other riders are home at that time?

Customers arrive at the automatic teller machine in accordance with a Poisson process with rate 12 per hour. The amount of money withdrawn on each transaction is a random variable with mean \(\$ 30\) and standard deviation \(\$ 50\). (A negative withdrawal means that money was deposited.) The machine is in use
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.