Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. 0 x < 0 +² F(x) = of 0≤x < 4 16 1 4 ≤ x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.) (a) Calculate P(X ≤ 3). (b) Calculate P(2.5 ≤ x ≤ 3). (c) Calculate P(X> 3.5). (d) What is the median checkout duration ? [solve 0.5 = F(ũ)]. (e) Obtain the density function f(x). f(x) = f'(x) (f) Calculate E(X). (g) Calculate V(X) and a V(x) ox 0≤x < 4 otherwise
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is the following. 0 x < 0 +² F(x) = of 0≤x < 4 16 1 4 ≤ x Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.) (a) Calculate P(X ≤ 3). (b) Calculate P(2.5 ≤ x ≤ 3). (c) Calculate P(X> 3.5). (d) What is the median checkout duration ? [solve 0.5 = F(ũ)]. (e) Obtain the density function f(x). f(x) = f'(x) (f) Calculate E(X). (g) Calculate V(X) and a V(x) ox 0≤x < 4 otherwise
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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Q2.2. Please answer g and h
![### Problem Statement and Exercises on Continuous Random Variables
Let \( X \) denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cumulative distribution function (CDF) is given by:
\[
F(x) = \begin{cases}
0 & x < 0 \\
x^2 & 0 \leq x < 4 \\
1 & 4 \leq x
\end{cases}
\]
Use the cumulative distribution function (CDF) to determine the following. (If necessary, round your answer to four decimal places.)
**(a)** Calculate \( P(X \leq 3) \).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(b)** Calculate \( P(2.5 \leq X \leq 3) \).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(c)** Calculate \( P(X > 3.5) \).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(d)** What is the median checkout duration \( \tilde{p} \)? (Solve \( 0.5 = F(\tilde{p}) \)).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(e)** Obtain the probability density function (PDF) \( f(x) \).
\[
f(X) = F'(x) \implies
f(x) = \begin{cases}
2x & 0 \leq x < 4 \\
0 & \text{otherwise}
\end{cases}
\]
**(f)** Calculate \( E(X) \) (Expected value of \( X \)).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(g)** Calculate \( V(X) \) and \( \sigma_x \) (Variance and Standard Deviation of \( X \)).
\[
V(X) \quad \text{Answer:} \quad \_\_\_\_\_
\sigma_x \quad \text{Answer:} \quad \_\_\_\_\_
\]
**(h)** If the borrower is charged an amount \( h(X) = X^2 \) when checkout duration is \( X \), compute the expected charge \(E(h](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6be9064-f0d8-4347-98bd-8b0b263612a8%2F4c3eed20-7826-4ab4-87ab-9ad30759dfd4%2F0g62wdi_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement and Exercises on Continuous Random Variables
Let \( X \) denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cumulative distribution function (CDF) is given by:
\[
F(x) = \begin{cases}
0 & x < 0 \\
x^2 & 0 \leq x < 4 \\
1 & 4 \leq x
\end{cases}
\]
Use the cumulative distribution function (CDF) to determine the following. (If necessary, round your answer to four decimal places.)
**(a)** Calculate \( P(X \leq 3) \).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(b)** Calculate \( P(2.5 \leq X \leq 3) \).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(c)** Calculate \( P(X > 3.5) \).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(d)** What is the median checkout duration \( \tilde{p} \)? (Solve \( 0.5 = F(\tilde{p}) \)).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(e)** Obtain the probability density function (PDF) \( f(x) \).
\[
f(X) = F'(x) \implies
f(x) = \begin{cases}
2x & 0 \leq x < 4 \\
0 & \text{otherwise}
\end{cases}
\]
**(f)** Calculate \( E(X) \) (Expected value of \( X \)).
\[
\text{Answer:} \quad \_\_\_\_\_
\]
**(g)** Calculate \( V(X) \) and \( \sigma_x \) (Variance and Standard Deviation of \( X \)).
\[
V(X) \quad \text{Answer:} \quad \_\_\_\_\_
\sigma_x \quad \text{Answer:} \quad \_\_\_\_\_
\]
**(h)** If the borrower is charged an amount \( h(X) = X^2 \) when checkout duration is \( X \), compute the expected charge \(E(h
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