(1) A continuous random variable W has p.d.f 2w, for 0
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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please answer d-g of the question
![**Problem (1)**
A continuous random variable \( W \) has a probability density function (p.d.f) given by:
\[
f(w) =
\begin{cases}
2w, & \text{for } 0 < w < 1 \\
0, & \text{otherwise}.
\end{cases}
\]
Given \( W = w \), a random variable \( T \) has the conditional p.d.f:
\[
f(t|w) =
\begin{cases}
\frac{1}{w}, & \text{for } 0 < t < w; \\
0, & \text{otherwise}.
\end{cases}
\]
Tasks:
(a) Find the joint p.d.f of \( W \) and \( T \), \( f(w, t) \).
(b) Find the marginal p.d.f. of \( T \).
(c) Find the mean of \( T \), \( \mu_T \).
(d) Find the variance of \( T \), \( \sigma_T^2 \).
(e) Given \( T = t \), find the conditional p.d.f. of \( W \), \( f(w|t) \).
(f) Find \( \mathbb{E}[W|t] \), for \( 0 < t < 1 \).
(g) Find \( \sigma^2_{W|t} \), for \( 0 < t < 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2Fb3c064fc-af7c-4adf-8daa-c9946aa9bdb2%2Fva3odi_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem (1)**
A continuous random variable \( W \) has a probability density function (p.d.f) given by:
\[
f(w) =
\begin{cases}
2w, & \text{for } 0 < w < 1 \\
0, & \text{otherwise}.
\end{cases}
\]
Given \( W = w \), a random variable \( T \) has the conditional p.d.f:
\[
f(t|w) =
\begin{cases}
\frac{1}{w}, & \text{for } 0 < t < w; \\
0, & \text{otherwise}.
\end{cases}
\]
Tasks:
(a) Find the joint p.d.f of \( W \) and \( T \), \( f(w, t) \).
(b) Find the marginal p.d.f. of \( T \).
(c) Find the mean of \( T \), \( \mu_T \).
(d) Find the variance of \( T \), \( \sigma_T^2 \).
(e) Given \( T = t \), find the conditional p.d.f. of \( W \), \( f(w|t) \).
(f) Find \( \mathbb{E}[W|t] \), for \( 0 < t < 1 \).
(g) Find \( \sigma^2_{W|t} \), for \( 0 < t < 1 \).
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