(d) What is the median checkout duration ? [solve 0.5 = F(M)]. (e) Obtain the density function f(x). f(x) = f'(x) (f) Calculate E(X). (g) Calculate V(X) and a. 0 < x < 4 otherwise
(d) What is the median checkout duration ? [solve 0.5 = F(M)]. (e) Obtain the density function f(x). f(x) = f'(x) (f) Calculate E(X). (g) Calculate V(X) and a. 0 < x < 4 otherwise
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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![**Text Transcription for Educational Website**
---
**Context:**
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 the following:
\[
F(x) =
\begin{cases}
0 & x < 0 \\
\frac{x^2}{16} & 0 \leq x < 4 \\
1 & 4 \leq x
\end{cases}
\]
**Tasks and Solutions:**
Use the cdf to obtain the following results. (If necessary, round your answer to four decimal places.)
(a) **Calculate \( P(X \leq 3) \).**
- Answer: 0.5625 ✅
(b) **Calculate \( P(2.5 \leq X \leq 3) \).**
- Answer: 0.171875 ✅
(c) **Calculate \( P(X > 3.5) \).**
- Answer: 0.234375 ✅
(d) **What is the median checkout duration \( \tilde{x} \)?** [Solve \( 0.5 = F(\tilde{x}) \).]
(e) **Obtain the density function \( f(x) \).**
\[
f(x) = F'(x) =
\begin{cases}
\text{(Expression for } \frac{d}{dx}(\frac{x^2}{16}) \text{)} & 0 \leq x < 4 \\
0 & \text{otherwise}
\end{cases}
\]
(f) **Calculate \( E(X) \).**
(g) **Calculate \( V(X) \) and \( \sigma_X \).**
- \( V(X) \) =
- \( \sigma_X \) =
(h) **If the borrower is charged an amount \( h(X) = X^2 \) when checkout duration is \( X \), compute the expected charge \( E[h(X)] \).**
---
Complete any remaining calculations as required to find the unknown values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F390df6bc-8765-4adb-882a-689888791ad2%2F8606bc0e-300e-463b-adda-5cf68ba693ee%2Fj3plnj_processed.png&w=3840&q=75)
Transcribed Image Text:**Text Transcription for Educational Website**
---
**Context:**
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 the following:
\[
F(x) =
\begin{cases}
0 & x < 0 \\
\frac{x^2}{16} & 0 \leq x < 4 \\
1 & 4 \leq x
\end{cases}
\]
**Tasks and Solutions:**
Use the cdf to obtain the following results. (If necessary, round your answer to four decimal places.)
(a) **Calculate \( P(X \leq 3) \).**
- Answer: 0.5625 ✅
(b) **Calculate \( P(2.5 \leq X \leq 3) \).**
- Answer: 0.171875 ✅
(c) **Calculate \( P(X > 3.5) \).**
- Answer: 0.234375 ✅
(d) **What is the median checkout duration \( \tilde{x} \)?** [Solve \( 0.5 = F(\tilde{x}) \).]
(e) **Obtain the density function \( f(x) \).**
\[
f(x) = F'(x) =
\begin{cases}
\text{(Expression for } \frac{d}{dx}(\frac{x^2}{16}) \text{)} & 0 \leq x < 4 \\
0 & \text{otherwise}
\end{cases}
\]
(f) **Calculate \( E(X) \).**
(g) **Calculate \( V(X) \) and \( \sigma_X \).**
- \( V(X) \) =
- \( \sigma_X \) =
(h) **If the borrower is charged an amount \( h(X) = X^2 \) when checkout duration is \( X \), compute the expected charge \( E[h(X)] \).**
---
Complete any remaining calculations as required to find the unknown values.
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