Problem 3) If Y is an Erlang (n-2, à-2) random variable, find the following: a) What is ELY? b) What is Var[Y]?
Problem 3) If Y is an Erlang (n-2, à-2) random variable, find the following: a) What is ELY? b) What is Var[Y]?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**EENG 3421 – Advanced Engineering Analysis**
**Exercise #8**
---
**Problem 1)** If \( Y \) is an exponential random variable with \( \text{Var}[Y] = 25 \), find the following:
a) What is the PDF of \( Y \)?
b) What is \( E[Y^2] \)?
c) What is \( P[Y > 5] \)?
**Problem 2)** If \( X \) is an Erlang \( (n, \lambda) \) random variable with parameter \( \lambda = 1/3 \) and expected value \( E[X] = 15 \), find the following:
a) What is the value of the parameter \( n \)?
b) What is the PDF of \( X \)?
c) What is \( \text{Var}[X] \)?
**Problem 3)** If \( Y \) is an Erlang \( (n = 2, \lambda = 2) \) random variable, find the following:
a) What is \( E[Y] \)?
b) What is \( \text{Var}[Y] \)?
c) Find \( P[0.5 \leq Y < 1.5] \).
**Problem 4)** If \( X \) is a continuous uniform \((-5, 5)\) random variable, find the following:
a) What is the PDF of \( X \)?
b) What is the CDF of \( X \)?
c) What is \( E[X^2] \)?
d) What is \( \text{Var}[X] \)?
e) What is \( E[e^X] \)?
**Problem 5)** If \( X \) is a continuous uniform random variable with expected value \( E[X] = 7 \) and variance \( \text{Var}[X] = 3 \), then what is the PDF of \( X \)?
**Problem 6)** Radars detect flying objects by measuring the power reflected from them. The reflected power of an aircraft can be modeled as a random variable \( Y \) with PDF:
\[
f_Y(y) =
\begin{cases}
\frac{1}{P_0} e^{-\frac{y}{P_0}}, & y \geq 0 \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe36fb7f7-6381-4add-b52b-9085dfd27844%2Fe958087a-7ceb-4ada-bd3f-47d8ac66d7d7%2Fu5me815_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**EENG 3421 – Advanced Engineering Analysis**
**Exercise #8**
---
**Problem 1)** If \( Y \) is an exponential random variable with \( \text{Var}[Y] = 25 \), find the following:
a) What is the PDF of \( Y \)?
b) What is \( E[Y^2] \)?
c) What is \( P[Y > 5] \)?
**Problem 2)** If \( X \) is an Erlang \( (n, \lambda) \) random variable with parameter \( \lambda = 1/3 \) and expected value \( E[X] = 15 \), find the following:
a) What is the value of the parameter \( n \)?
b) What is the PDF of \( X \)?
c) What is \( \text{Var}[X] \)?
**Problem 3)** If \( Y \) is an Erlang \( (n = 2, \lambda = 2) \) random variable, find the following:
a) What is \( E[Y] \)?
b) What is \( \text{Var}[Y] \)?
c) Find \( P[0.5 \leq Y < 1.5] \).
**Problem 4)** If \( X \) is a continuous uniform \((-5, 5)\) random variable, find the following:
a) What is the PDF of \( X \)?
b) What is the CDF of \( X \)?
c) What is \( E[X^2] \)?
d) What is \( \text{Var}[X] \)?
e) What is \( E[e^X] \)?
**Problem 5)** If \( X \) is a continuous uniform random variable with expected value \( E[X] = 7 \) and variance \( \text{Var}[X] = 3 \), then what is the PDF of \( X \)?
**Problem 6)** Radars detect flying objects by measuring the power reflected from them. The reflected power of an aircraft can be modeled as a random variable \( Y \) with PDF:
\[
f_Y(y) =
\begin{cases}
\frac{1}{P_0} e^{-\frac{y}{P_0}}, & y \geq 0 \
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