Problem 4) Continuous random variable X has the following PDF: -1≤x≤3 fx(x) = otherwise = Define the random variable Y by Y a) Find E[X] and Var[X]. b) Find h(E[X]) and E[h(X)]. c) Find E[Y] and Var[Y]. h(x) = x²
Problem 4) Continuous random variable X has the following PDF: -1≤x≤3 fx(x) = otherwise = Define the random variable Y by Y a) Find E[X] and Var[X]. b) Find h(E[X]) and E[h(X)]. c) Find E[Y] and Var[Y]. h(x) = x²
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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![**Problem 1**: The cumulative distribution function of random variable \( X \) is:
\[
F_X(x) =
\begin{cases}
0 & x < -1 \\
\frac{x+1}{2} & -1 \leq x < 1 \\
1 & x \geq 1
\end{cases}
\]
a) Find \( P[X > 1/2] \).
b) Find \( P[-1/2 < X \leq 3/4] \).
c) Find \( P[|X| \leq 1/2] \).
d) What is the value of \( a \) such that \( P[X \leq a] = 0.8 \).
e) Find the PDF \( f_X(x) \) of \( X \).
---
**Problem 2**: The CDF of random variable \( W \) is:
\[
F_W(w) =
\begin{cases}
0 & x < -5 \\
\frac{w+5}{8} & -5 \leq w < -3 \\
\frac{1}{4} & -3 \leq w < 3 \\
\frac{1}{4} + \frac{3(w-3)}{8} & 3 \leq w < 5 \\
1 & w \geq 5
\end{cases}
\]
a) Find \( P[W \leq 4] \).
b) Find \( P[-2 < W \leq 2] \).
c) Find \( P[W > 0] \).
d) What is the value of \( a \) such that \( P[W \leq a] = 0.5 \).
---
**Problem 3**: The random variable \( X \) has the following probability density function:
\[
f_X(x) =
\begin{cases}
cx & 0 \leq x \leq 2 \\
0 & \text{otherwise}
\end{cases}
\]
a) Find the constant \( c \).
b) Find \( P[0 \leq X \leq 1] \).
c) Find \( P[-1/2 \leq X \leq 1/2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe36fb7f7-6381-4add-b52b-9085dfd27844%2F53b35274-5a03-4cab-8b8e-077f60797372%2Fgwiycx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1**: The cumulative distribution function of random variable \( X \) is:
\[
F_X(x) =
\begin{cases}
0 & x < -1 \\
\frac{x+1}{2} & -1 \leq x < 1 \\
1 & x \geq 1
\end{cases}
\]
a) Find \( P[X > 1/2] \).
b) Find \( P[-1/2 < X \leq 3/4] \).
c) Find \( P[|X| \leq 1/2] \).
d) What is the value of \( a \) such that \( P[X \leq a] = 0.8 \).
e) Find the PDF \( f_X(x) \) of \( X \).
---
**Problem 2**: The CDF of random variable \( W \) is:
\[
F_W(w) =
\begin{cases}
0 & x < -5 \\
\frac{w+5}{8} & -5 \leq w < -3 \\
\frac{1}{4} & -3 \leq w < 3 \\
\frac{1}{4} + \frac{3(w-3)}{8} & 3 \leq w < 5 \\
1 & w \geq 5
\end{cases}
\]
a) Find \( P[W \leq 4] \).
b) Find \( P[-2 < W \leq 2] \).
c) Find \( P[W > 0] \).
d) What is the value of \( a \) such that \( P[W \leq a] = 0.5 \).
---
**Problem 3**: The random variable \( X \) has the following probability density function:
\[
f_X(x) =
\begin{cases}
cx & 0 \leq x \leq 2 \\
0 & \text{otherwise}
\end{cases}
\]
a) Find the constant \( c \).
b) Find \( P[0 \leq X \leq 1] \).
c) Find \( P[-1/2 \leq X \leq 1/2
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