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A random variable has density function f(x) = 1.6 - 1.2x, for 0≤x≤1. a) Calculate the variance of X. Var (X) = = 0.073 X

A random variable has density function f(x) = 1.6 - 1.2x, for 0≤x≤1. a) Calculate the variance of X. Var (X) = = 0.073 X

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Transcription and Explanation for Educational Website** --- ### Problem Statement A random variable has density function \( f(x) = 1.6 - 1.2x \), for \( 0 \leq x \leq 1 \). #### a) Calculate the variance of \( X \). \[ \text{Var}(X) = 0.073 \; \text{\red{✗}} \] The calculation shown for the variance of \( X \) is marked incorrect. #### b) Calculate the variance of \( g(X) = 3X + 1 \). \[ \text{Var}(g(X)) = 0.657 \; \text{\red{✗}} \] The calculation for the variance of \( g(X) \) is also marked incorrect. --- ### Explanation This image presents a problem related to calculating variances for a random variable characterized by a given probability density function (PDF). The function is defined on the interval \( [0, 1] \). **Density Function:** - \( f(x) = 1.6 - 1.2x \) is the PDF over the specified range. **Tasks:** 1. **Variance of \( X \):** - Calculate the variance using the given density function. 2. **Variance of \( g(X) = 3X + 1 \):** - Apply transformation properties for variance: - If \( g(X) = aX + b \), then \( \text{Var}(g(X)) = a^2 \text{Var}(X) \). Both answers provided for \( \text{Var}(X) \) and \( \text{Var}(g(X)) \) were incorrect, which suggests a reevaluation of the integration and transformation steps is needed. **Note:** Calculating the correct variance involves integrating the square of deviation from the mean weighted by the density function.
Transcribed Image Text:**Transcription and Explanation for Educational Website** --- ### Problem Statement A random variable has density function \( f(x) = 1.6 - 1.2x \), for \( 0 \leq x \leq 1 \). #### a) Calculate the variance of \( X \). \[ \text{Var}(X) = 0.073 \; \text{\red{✗}} \] The calculation shown for the variance of \( X \) is marked incorrect. #### b) Calculate the variance of \( g(X) = 3X + 1 \). \[ \text{Var}(g(X)) = 0.657 \; \text{\red{✗}} \] The calculation for the variance of \( g(X) \) is also marked incorrect. --- ### Explanation This image presents a problem related to calculating variances for a random variable characterized by a given probability density function (PDF). The function is defined on the interval \( [0, 1] \). **Density Function:** - \( f(x) = 1.6 - 1.2x \) is the PDF over the specified range. **Tasks:** 1. **Variance of \( X \):** - Calculate the variance using the given density function. 2. **Variance of \( g(X) = 3X + 1 \):** - Apply transformation properties for variance: - If \( g(X) = aX + b \), then \( \text{Var}(g(X)) = a^2 \text{Var}(X) \). Both answers provided for \( \text{Var}(X) \) and \( \text{Var}(g(X)) \) were incorrect, which suggests a reevaluation of the integration and transformation steps is needed. **Note:** Calculating the correct variance involves integrating the square of deviation from the mean weighted by the density function.
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Step 1: Write the given information.

Given information:

f open parentheses x close parentheses equals 1.6 minus 1.2 x comma space 0 less or equal than x less or equal than 1

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