Properties of expected value and variance of a continuous random variable. a)Assume that X is a random variable and a and b are constant. With the integral definition of the expected value of a continuous random variable show that E(aX+b)=aE(x)+b and conclude like the theoretical questions of HW 3 , 2a) that V(aX + b) € b)Also using the fact that E(x) is a constant show that . V(X ) V(X) = E[(X – µ)³] = ÈX²) – [E(X)²

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
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Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 10E
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### Properties of Expected Value and Variance of a Continuous Random Variable

#### 1. Problem Statement
Consider the following properties related to expected value and variance:

**a)** Assume that \( X \) is a random variable and \( a \) and \( b \) are constants. Using the integral definition of the expected value for a continuous random variable, demonstrate that:

\[
E(aX + b) = aE(X) + b
\]

Conclude in a manner similar to the theoretical questions in Homework 3, question 2a, that:

\[
V(aX + b) = a^2 \cdot V(X)
\]

**b)** Additionally, given that \( E(X) \) is a constant, prove that:

\[
V(X) = E[(X - \mu)^2] = E[X^2] - [E(X)]^2
\]

In this proof, details are crucial to understanding the effect of linear transformations on expected values and variances, alongside the formula for calculating variance directly from the expected value of squared deviations.
Transcribed Image Text:### Properties of Expected Value and Variance of a Continuous Random Variable #### 1. Problem Statement Consider the following properties related to expected value and variance: **a)** Assume that \( X \) is a random variable and \( a \) and \( b \) are constants. Using the integral definition of the expected value for a continuous random variable, demonstrate that: \[ E(aX + b) = aE(X) + b \] Conclude in a manner similar to the theoretical questions in Homework 3, question 2a, that: \[ V(aX + b) = a^2 \cdot V(X) \] **b)** Additionally, given that \( E(X) \) is a constant, prove that: \[ V(X) = E[(X - \mu)^2] = E[X^2] - [E(X)]^2 \] In this proof, details are crucial to understanding the effect of linear transformations on expected values and variances, alongside the formula for calculating variance directly from the expected value of squared deviations.
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