7. Given X~U(0,22), complete the diagram and calculate the probabilities: f(x) а. Р(х < 20) b. Р(x > 6) c. P(10 < x s 15) d. P(x > 10|x > 6)
7. Given X~U(0,22), complete the diagram and calculate the probabilities: f(x) а. Р(х < 20) b. Р(x > 6) c. P(10 < x s 15) d. P(x > 10|x > 6)
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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Question
![**Exercise 7: Probability and Uniform Distribution**
Given that \( X \sim U(0,22) \), complete the diagram and calculate the probabilities:
The diagram is a horizontal line representing the probability density function (pdf) of a uniform distribution from 0 to 22. The height of the line is constant at 1/22, indicating the uniform distribution's probability density. The x-axis is labeled as \( x \), and the y-axis is labeled as \( f(x) \).
To determine the probabilities, use the uniform distribution properties where the probability for any interval \([a, b]\) within \([0, 22]\) is calculated as:
\[ P(a \leq X \leq b) = \frac{b-a}{22} \]
**Questions:**
a. \( P(x < 20) \)
b. \( P(x > 6) \)
c. \( P(10 \leq x \leq 15) \)
d. \( P(x > 10 \mid x > 6) \)
**Solutions:**
a. **\( P(x < 20) \)**:
Calculate the probability from 0 to 20.
\[ P(x < 20) = \frac{20 - 0}{22} = \frac{20}{22} = \frac{10}{11} \]
b. **\( P(x > 6) \)**:
Calculate the probability from 6 to 22.
\[ P(x > 6) = \frac{22 - 6}{22} = \frac{16}{22} = \frac{8}{11} \]
c. **\( P(10 \leq x \leq 15) \)**:
Calculate the probability from 10 to 15.
\[ P(10 \leq x \leq 15) = \frac{15 - 10}{22} = \frac{5}{22} \]
d. **\( P(x > 10 \mid x > 6) \)**:
Calculate the conditional probability given \( x > 6 \).
\[ P(x > 10 \mid x > 6) = \frac{P(x > 10 \cap x > 6)}{P(x > 6)} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4726056b-a0d8-4167-a534-8f03670a441a%2Fe6b8f6dd-7635-46d7-b54b-cd360949aa37%2F4q04eol_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 7: Probability and Uniform Distribution**
Given that \( X \sim U(0,22) \), complete the diagram and calculate the probabilities:
The diagram is a horizontal line representing the probability density function (pdf) of a uniform distribution from 0 to 22. The height of the line is constant at 1/22, indicating the uniform distribution's probability density. The x-axis is labeled as \( x \), and the y-axis is labeled as \( f(x) \).
To determine the probabilities, use the uniform distribution properties where the probability for any interval \([a, b]\) within \([0, 22]\) is calculated as:
\[ P(a \leq X \leq b) = \frac{b-a}{22} \]
**Questions:**
a. \( P(x < 20) \)
b. \( P(x > 6) \)
c. \( P(10 \leq x \leq 15) \)
d. \( P(x > 10 \mid x > 6) \)
**Solutions:**
a. **\( P(x < 20) \)**:
Calculate the probability from 0 to 20.
\[ P(x < 20) = \frac{20 - 0}{22} = \frac{20}{22} = \frac{10}{11} \]
b. **\( P(x > 6) \)**:
Calculate the probability from 6 to 22.
\[ P(x > 6) = \frac{22 - 6}{22} = \frac{16}{22} = \frac{8}{11} \]
c. **\( P(10 \leq x \leq 15) \)**:
Calculate the probability from 10 to 15.
\[ P(10 \leq x \leq 15) = \frac{15 - 10}{22} = \frac{5}{22} \]
d. **\( P(x > 10 \mid x > 6) \)**:
Calculate the conditional probability given \( x > 6 \).
\[ P(x > 10 \mid x > 6) = \frac{P(x > 10 \cap x > 6)}{P(x > 6)} =
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