(b) What is P(X ≤ 0.7) [i.e., F(0.7)]? (Round your answer to four decimal places.) 0.1960 (c) Using the cdf from (a), what is P(0.45 < x≤ 0.7)? (Round your answer to four decimal places.) 0.1869 What is P(0.45 ≤ x ≤ 0.7)? (Round your answer to four decimal places.) 0.1869 (d) What is the 75th percentile of the distribution? (Round your answer to four decimal places.) 7 0.4207 x (e) Compute E(X) and ox. (Round your answers to four decimal places.) E(X)= 0.8 0.1206 (f) What is the probability that X is more than 1 standard deviation from its mean value? (Round your answer to four decimal places.) 10 0.1551 x
(b) What is P(X ≤ 0.7) [i.e., F(0.7)]? (Round your answer to four decimal places.) 0.1960 (c) Using the cdf from (a), what is P(0.45 < x≤ 0.7)? (Round your answer to four decimal places.) 0.1869 What is P(0.45 ≤ x ≤ 0.7)? (Round your answer to four decimal places.) 0.1869 (d) What is the 75th percentile of the distribution? (Round your answer to four decimal places.) 7 0.4207 x (e) Compute E(X) and ox. (Round your answers to four decimal places.) E(X)= 0.8 0.1206 (f) What is the probability that X is more than 1 standard deviation from its mean value? (Round your answer to four decimal places.) 10 0.1551 x
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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![## Probability Density Function (PDF) and Cumulative Distribution Function (CDF)
### PDF Explanation
Let \( X \) denote the amount of space occupied by an article in a 1-t\(^2\) packing container. The probability density function (pdf) of \( X \) is defined as:
\[ f(x) =
\begin{cases}
12x^2(1-x) & \text{for } 0 < x < 1\\
0 & \text{otherwise}
\end{cases}
\]
### Graphing the PDF
The graphs shown for the PDF \( f(x) \) illustrate the function over the interval \( 0 < x < 1 \). Various graph shapes are proposed, and the correct graph is indicated with a check mark (✓). This graph shows a peak within the range and zero outside \( 0 < x < 1 \).
### CDF Explanation
The cumulative distribution function (CDF) \( F(x) \) is derived as follows:
\[ F(x) =
\begin{cases}
0 & \text{for } x < 0\\
-8x^3 + 9x^2 & \text{for } 0 \leq x \leq 1\\
1 & \text{for } x > 1
\end{cases}
\]
### Graphing the CDF
Various graphs propose the shape for \( F(x) \). The correct graph, marked with a check (✗ indicates incorrect), shows a gradual increase from 0 to 1 over the interval \( 0 \leq x \leq 1 \) and remains constant outside this interval.
### Questions and Answers
(b) **Probability \( P(X \leq 0.7) \):** Given as \( F(0.7) \), the answer is **0.9600**.
(c) **Probability \( P(0.45 < X \leq 0.7) \):** Using the CDF, the answer is **0.1889**.
(d) **Probability \( P(0.45 \leq X \leq 0.7) \):** The given result matches (c): **0.1889**.
(e) **75th Percentile of the Distribution:** The answer is **0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F913a0921-d398-48e9-845b-58c3dce12c9b%2Fd9cb2bca-e88b-42dd-b090-acb3a21e53c7%2Fy1h6id_processed.png&w=3840&q=75)
Transcribed Image Text:## Probability Density Function (PDF) and Cumulative Distribution Function (CDF)
### PDF Explanation
Let \( X \) denote the amount of space occupied by an article in a 1-t\(^2\) packing container. The probability density function (pdf) of \( X \) is defined as:
\[ f(x) =
\begin{cases}
12x^2(1-x) & \text{for } 0 < x < 1\\
0 & \text{otherwise}
\end{cases}
\]
### Graphing the PDF
The graphs shown for the PDF \( f(x) \) illustrate the function over the interval \( 0 < x < 1 \). Various graph shapes are proposed, and the correct graph is indicated with a check mark (✓). This graph shows a peak within the range and zero outside \( 0 < x < 1 \).
### CDF Explanation
The cumulative distribution function (CDF) \( F(x) \) is derived as follows:
\[ F(x) =
\begin{cases}
0 & \text{for } x < 0\\
-8x^3 + 9x^2 & \text{for } 0 \leq x \leq 1\\
1 & \text{for } x > 1
\end{cases}
\]
### Graphing the CDF
Various graphs propose the shape for \( F(x) \). The correct graph, marked with a check (✗ indicates incorrect), shows a gradual increase from 0 to 1 over the interval \( 0 \leq x \leq 1 \) and remains constant outside this interval.
### Questions and Answers
(b) **Probability \( P(X \leq 0.7) \):** Given as \( F(0.7) \), the answer is **0.9600**.
(c) **Probability \( P(0.45 < X \leq 0.7) \):** Using the CDF, the answer is **0.1889**.
(d) **Probability \( P(0.45 \leq X \leq 0.7) \):** The given result matches (c): **0.1889**.
(e) **75th Percentile of the Distribution:** The answer is **0.
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