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 P-f x² F(x) 0≤x < 4 16 1 4 SX Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.) (a) Calculate P(X ≤ 3). 0.5625 (b) Calculate P(2.5 ≤ x ≤ 3). 0.1719 (c) Calculate P(X> 3.5). 0.2344 (d) What is the median checkout duration ? [solve 0.5 = F(μ)]. (e) Obtain the density function f(x). f(x) = F'(x) 0≤x < 4 otherwise (f) Calculate E(X). (g) Calculate V(X) and a V(X) %% (h) If the borrower is charged an amount h(x) = x² when checkout duration is X, compute the expected charge E[h(X)].
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 P-f x² F(x) 0≤x < 4 16 1 4 SX Use the cdf to obtain the following. (If necessary, round your answer to four decimal places.) (a) Calculate P(X ≤ 3). 0.5625 (b) Calculate P(2.5 ≤ x ≤ 3). 0.1719 (c) Calculate P(X> 3.5). 0.2344 (d) What is the median checkout duration ? [solve 0.5 = F(μ)]. (e) Obtain the density function f(x). f(x) = F'(x) 0≤x < 4 otherwise (f) Calculate E(X). (g) Calculate V(X) and a V(X) %% (h) If the borrower is charged an amount h(x) = x² when checkout duration is X, compute the expected charge E[h(X)].
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 19E
Related questions
Question
P17. Please only answer the questions to the left of the red X. Thank you
![### Probability and Statistics: Continuous Random Variables
#### Distribution Functions and Calculations
Let \( X \) denote the amount of time a book on two-hour reserve is actually checked out. Suppose the cumulative distribution function (CDF) is given by:
\[ F(x) =
\begin{cases}
0 & x < 0 \\
\frac{x^2}{16} & 0 \leq x < 4 \\
1 & 4 \leq x
\end{cases}
\]
Using this CDF, we can derive various properties and statistics of the random variable \( X \).
#### (a) Calculate \( P(X \leq 3) \).
To find \( P(X \leq 3) \), we evaluate the CDF at \( x = 3 \):
\[ F(3) = \frac{3^2}{16} = \frac{9}{16} \approx 0.5625 \]
#### (b) Calculate \( P(2.5 \leq X \leq 3) \).
This probability is given by the difference \( F(3) - F(2.5) \):
\[ F(2.5) = \frac{2.5^2}{16} = \frac{6.25}{16} \approx 0.3906 \]
\[ P(2.5 \leq X \leq 3) = F(3) - F(2.5) = 0.5625 - 0.3906 \approx 0.1719 \]
#### (c) Calculate \( P(X > 3.5) \).
This probability is given by \( 1 - F(3.5) \):
\[ F(3.5) = \frac{3.5^2}{16} = \frac{12.25}{16} \approx 0.7656 \]
\[ P(X > 3.5) = 1 - 0.7656 \approx 0.2344 \]
#### (d) What is the median checkout duration \( \tilde{x} \)?
The median checkout duration is the value \( \tilde{x} \) such that \( F(\tilde{x}) = 0.5 \):
\[ F(\tilde{x}) = \frac{\tilde{x}^2}{16} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda5a8041-aeed-45a6-a7fc-c4165c2d9a1f%2Fa9ec5c42-435a-48cd-a4ce-d16d0a4e14a3%2Foiplsxp_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability and Statistics: Continuous Random Variables
#### Distribution Functions and Calculations
Let \( X \) denote the amount of time a book on two-hour reserve is actually checked out. Suppose the cumulative distribution function (CDF) is given by:
\[ F(x) =
\begin{cases}
0 & x < 0 \\
\frac{x^2}{16} & 0 \leq x < 4 \\
1 & 4 \leq x
\end{cases}
\]
Using this CDF, we can derive various properties and statistics of the random variable \( X \).
#### (a) Calculate \( P(X \leq 3) \).
To find \( P(X \leq 3) \), we evaluate the CDF at \( x = 3 \):
\[ F(3) = \frac{3^2}{16} = \frac{9}{16} \approx 0.5625 \]
#### (b) Calculate \( P(2.5 \leq X \leq 3) \).
This probability is given by the difference \( F(3) - F(2.5) \):
\[ F(2.5) = \frac{2.5^2}{16} = \frac{6.25}{16} \approx 0.3906 \]
\[ P(2.5 \leq X \leq 3) = F(3) - F(2.5) = 0.5625 - 0.3906 \approx 0.1719 \]
#### (c) Calculate \( P(X > 3.5) \).
This probability is given by \( 1 - F(3.5) \):
\[ F(3.5) = \frac{3.5^2}{16} = \frac{12.25}{16} \approx 0.7656 \]
\[ P(X > 3.5) = 1 - 0.7656 \approx 0.2344 \]
#### (d) What is the median checkout duration \( \tilde{x} \)?
The median checkout duration is the value \( \tilde{x} \) such that \( F(\tilde{x}) = 0.5 \):
\[ F(\tilde{x}) = \frac{\tilde{x}^2}{16} =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
