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Let X have a distribution with pdf Find (a) Mean of X. f(x)= - X 2 1 < x < e. (c) Pr(1 < X < 2). If X is an exponential random variable with parameter λ = 1, find fy (y) (prob. density function) of the random variable Y, which is defined by Y = ex. (b) Median X. (hint) fx(x)=e" and Fx(x) = 1-e. Find Fy (y) first, and get fy (y) by taking a derivative for Fy (y).

Let X have a distribution with pdf Find (a) Mean of X. f(x)= - X 2 1 < x < e. (c) Pr(1 < X < 2). If X is an exponential random variable with parameter λ = 1, find fy (y) (prob. density function) of the random variable Y, which is defined by Y = ex. (b) Median X. (hint) fx(x)=e" and Fx(x) = 1-e. Find Fy (y) first, and get fy (y) by taking a derivative for Fy (y).

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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**Title: Probability Distribution Exercise** --- **Problem Statement:** Let \( X \) have a distribution with the probability density function (pdf) \[ f(x) = \frac{1}{x}, \quad 1 < x < e. \] **Tasks:** 1. **Find the Mean of \( X \).** 2. **Find the Median of \( X \).** 3. **Calculate \( \Pr(1 < X < 2) \).** --- **Additional Problem:** If \( X \) is an exponential random variable with parameter \( \lambda = 1 \), determine the probability density function \( f_Y(y) \) of the random variable \( Y \), defined by \( Y = e^X \). - **Hint:** - The pdf of \( X \) is \( f_X(x) = e^{-x} \). - The cumulative distribution function (CDF) is \( F_X(x) = 1 - e^{-x} \). **Steps:** 1. Find \( F_Y(y) \) first. 2. Obtain \( f_Y(y) \) by taking the derivative of \( F_Y(y) \). ---
Transcribed Image Text:**Title: Probability Distribution Exercise** --- **Problem Statement:** Let \( X \) have a distribution with the probability density function (pdf) \[ f(x) = \frac{1}{x}, \quad 1 < x < e. \] **Tasks:** 1. **Find the Mean of \( X \).** 2. **Find the Median of \( X \).** 3. **Calculate \( \Pr(1 < X < 2) \).** --- **Additional Problem:** If \( X \) is an exponential random variable with parameter \( \lambda = 1 \), determine the probability density function \( f_Y(y) \) of the random variable \( Y \), defined by \( Y = e^X \). - **Hint:** - The pdf of \( X \) is \( f_X(x) = e^{-x} \). - The cumulative distribution function (CDF) is \( F_X(x) = 1 - e^{-x} \). **Steps:** 1. Find \( F_Y(y) \) first. 2. Obtain \( f_Y(y) \) by taking the derivative of \( F_Y(y) \). ---
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