/*! 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 24 Suppose \(f\) is a probability d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 24

Suppose \(f\) is a probability density function for the random variable \(X\) with mean \(\mu .\) Show that its variance satisfies $$ \operatorname{Var}(X)=\int_{-\infty}^{\infty} X^{2} f(X) d X-\mu^{2} $$

Short Answer

Expert verified
\(\operatorname{Var}(X) = \int_{-\infty}^{\infty} X^2 f(X) \, dX - \mu^2\).

Step by step solution

01

Understanding Variance Definition

The variance of a random variable, denoted as \(\operatorname{Var}(X)\), measures how much the values of the random variable differ from the mean. It is defined as \(\operatorname{Var}(X) = E[(X - \mu)^2]\), where \(E\) represents the expected value and \(\mu\) is the mean.
02

Expand the Variance Expression

Expand the expression \(E[(X - \mu)^2]\) using the formula \((a-b)^2 = a^2 - 2ab + b^2\). This gives us: \[E[X^2 - 2\mu X + \mu^2] = E[X^2] - 2\mu E[X] + E[\mu^2]\].
03

Break Down Each Expectation Term

Since \(\mu\) is a constant, \(E[\mu^2] = \mu^2\) and \(E[X]\) is simply \(\mu\) (because \(f\) is the probability density function for the random variable with mean \(\mu\)). Therefore, the expression simplifies to \(E[X^2] - 2\mu^2 + \mu^2 = E[X^2] - \mu^2\).
04

Calculate \(E[X^2]\) Using an Integral

The expected value \(E[X^2]\) for a continuous random variable with density function \(f(X)\) is given by the integral \(\int_{-\infty}^{\infty} X^2 f(X) \, dX\). Thus, \(E[X^2] = \int_{-\infty}^{\infty} X^2 f(X) \, dX\).
05

Final Step of Variance Calculation

Substitute \(E[X^2]\) back into the variance expression: \[\operatorname{Var}(X) = \int_{-\infty}^{\infty} X^2 f(X) \, dX - \mu^2\]. This shows that the variance of \(X\) can be represented as the expected value of \(X^2\) minus the square of the mean.

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.

Probability Density Function
A probability density function (PDF) is a crucial concept in understanding continuous random variables. It describes how probability is distributed over the values of the random variable. Unlike discrete random variables, continuous variables can take an infinite number of values, and thus, we use a PDF instead of a probability mass function.

A PDF, denoted by a function like \(f(X)\), assigns probabilities to intervals, rather than specific values. It ensures that the probability of the random variable falling within a particular interval is found by integrating the PDF over that interval.

Key aspects of a PDF include:
  • The total area under the curve of the PDF over the entire range is 1, representing the total probability.
  • Unlike simple probabilities, the value of the PDF at a specific point can be greater than 1, since it represents a density not probability.
  • To find the probability that a random variable \(X\) falls between values \(a\) and \(b\), one computes \(\int_{a}^{b} f(x)\, dx\).
Understanding a probability density function is essential for calculating expectations and variances of continuous random variables.
Expected Value
The expected value, or mean, of a continuous random variable is a central concept in probability and statistics. It provides the average or "expected" outcome of a random process based on its probability distribution.

For a continuous random variable \(X\) with a probability density function \(f(X)\), the expected value, denoted as \(E[X]\), is calculated using the integral:
  • \(E[X] = \int_{-\infty}^{\infty} X f(X) \, dX\)
This integral sums up all outcomes, weighted by their likelihood, provided through the density function.

Some properties of expected value include:
  • Linearity: \(E[aX + b] = aE[X] + b\), meaning you can factor constants out of the expectation.
  • Indicator of central tendency: It tells us where the central mass of the distribution is located.
Understanding expected value is crucial for determining other moments like variance, which measures the spread around the mean.
Continuous Random Variable
Continuous random variables are a core concept of probability theory. Unlike discrete variables, which take on countable values, continuous variables can assume any value within a range. This feature makes them ideal for modeling scenarios where precision and possible value ranges are infinite, such as measuring time, height, or temperature.

Defining a continuous random variable requires a probability density function (PDF). The PDF characterizes the distribution of the variable and allows us to calculate probabilities over intervals.

Key characteristics of continuous random variables include:
  • The probability that a continuous random variable takes an exact value is zero. Instead, we calculate the probability over intervals.
  • The calculation involves integrating over the PDF, employing methods like finding expected values and variances.
These properties enable continuous random variables to model real-world phenomena accurately, capturing an endless number of potential outcomes across any given domain.

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

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{2}^{\infty} \frac{1}{\ln x} d x$$

Use a CAS to perform the integrations. Evaluate the integrals a. \(\int x \ln x d x \quad\) b. \(\int x^{2} \ln x d x \quad\) c. \(\int x^{3} \ln x d x.\) d. What pattern do you see? Predict the formula for \(\int x^{4} \ln x d x\) and then see if you are correct by evaluating it with a CAS. e. What is the formula for \(\int x^{n} \ln x d x, n \geq 1 ?\) Check your answer using a CAS.

Life expectancy At birth, a French citizen has an average life expectancy of 81 years with a standard deviation of 7 years. If 100 newly born French babies are selected at random, how many would you expert to live between 75 and 85 years? Assume life expectancy is normally distributed.

Three people are asked their opinion in a poll about a particular brand of a common product found in grocery stores. They can answer in one of three ways: "Like the product brand" (L), "Dislike the product brand" (D), or "Undecided" (U). For each outcome, the random variable \(X\) assigns the number of L's that appear. $$ \begin{array}{l}{\text { a. Find the set of possible outcomes and the range of } X .} \\ {\text { b. Create a probability bar graph for } X \text { . }} \\\ {\text { c. What is the probability that at least two people like the product }} \\ {\text { brand? }} \\ {\text { d. What is the probability that no more than one person dislikes }} \\ {\text { the product brand? }}\end{array} $$

Prove that the sum \(T\) in the Trapezoidal Rule for \(\int_{a}^{b} f(x) d x\) is a Riemann sum for \(f\) continuous on \([a, b] .\) Hint: Use the Intermediate Value Theorem to show the existence of \(c_{k}\) in the subinterval \(\left[x_{k-1}, x_{k}\right]\) satisfying \(f\left(c_{k}\right)=\left(f\left(x_{k-1}\right)+f\left(x_{k}\right)\right) / 2 . )\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.