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Let X be an exponentially distributed random variable with parameter = 1. (recall that if exponential (A), then Fx (x) = max(0,1 - exp(-2x)). X eax, where a is a real number. 2 Let Ya a. Find the cdf of Ya (Hint: the cdf has the form max(0,1-px9). Find the constants p, q.) b. Find the pdf of Ya c. Find the expected value of Ya. (Hint: there are two cases). d. Find the variance of Ya. (Hint: there are two cases).

Let X be an exponentially distributed random variable with parameter = 1. (recall that if exponential (A), then Fx (x) = max(0,1 - exp(-2x)). X eax, where a is a real number. 2 Let Ya a. Find the cdf of Ya (Hint: the cdf has the form max(0,1-px9). Find the constants p, q.) b. Find the pdf of Ya c. Find the expected value of Ya. (Hint: there are two cases). d. Find the variance of Ya. (Hint: there are two cases).

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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Please do part a-d and show your work
Let \( X \) be an exponentially distributed random variable with parameter \( \lambda = 1 \). (Recall that if \( X \sim \text{exponential}(\lambda) \), then \( F_X(x) = \max(0, 1 - \exp(-\lambda x)) \). Let \( Y_a = e^{aX} \), where \( a \) is a real number. a. Find the CDF of \( Y_a \). (Hint: the CDF has the form \( \max(0, 1 - px^q) \). Find the constants \( p, q \)). b. Find the PDF of \( Y_a \). c. Find the expected value of \( Y_a \). (Hint: there are two cases). d. Find the variance of \( Y_a \). (Hint: there are two cases).
Transcribed Image Text:Let \( X \) be an exponentially distributed random variable with parameter \( \lambda = 1 \). (Recall that if \( X \sim \text{exponential}(\lambda) \), then \( F_X(x) = \max(0, 1 - \exp(-\lambda x)) \). Let \( Y_a = e^{aX} \), where \( a \) is a real number. a. Find the CDF of \( Y_a \). (Hint: the CDF has the form \( \max(0, 1 - px^q) \). Find the constants \( p, q \)). b. Find the PDF of \( Y_a \). c. Find the expected value of \( Y_a \). (Hint: there are two cases). d. Find the variance of \( Y_a \). (Hint: there are two cases).
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Step 1: pdf and cdf of a random variable obtained.

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