fx.v (x,y) = (x + y²), 0
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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Please answer the questions included in the image.
![### Transcription for Educational Purposes
**Joint Probability Density Function:**
\[ f_{X,Y}(x,y) = \frac{6}{5} (x + y^2), \quad 0 \le x \le 1, \, 0 \le y \le 1 \]
**Questions:**
1. **a)** What is the marginal probability density function (pdf) of \( f_Y(y) \)?
2. **b)** What is the conditional pdf of \( f_{X|Y}(x|y = 0.5) \)?
**Q3. Exponential Distribution Properties:**
The Exponential Distribution has a *memoryless* property. Intuitively, this means that the probability of a customer service answering your call (assuming waiting time is exponential) in the next 10 minutes is the same, regardless of whether you have waited an hour on the line or just picked up the phone. Formally, if \( X \sim \text{exponential}(\lambda) \), then \( f(x) = \lambda \exp(-\lambda x) \). Let \( t \) and \( s \) be two positive numbers. Use the definition of conditional probability to show:
\[ P(X > t + s \mid X > t) = P(X > s) \]
*Hint:* Find the cumulative distribution function (cdf) of \( X \) first, and note that \( P(X > t + s \cap X > t) = P(X > t + s) \).
**Q4. Fair Die Game:**
Roll a fair die (uniform distribution over 1, 2, 3, 4, 5, 6) repeatedly. You and Peter are betting on the number shown on each roll.
- If the number is 4 or less, you win $1; otherwise, you pay Peter $2.5.
- **a)** What is the expected value of the payoff for you?
- **b)** What is the variance of your payoff?
**Q5. Uniform Distribution:**
Let \( X \) be uniformly distributed (continuous) on the interval \([1,2]\). Find \( E(1/X) \).
**Q6. Exponential Distribution for Light Bulbs:**
Assume the lifetime of light bulbs follows an exponential distribution with pdf:
\[ f(x) = \lambda \exp(-\lambda x) \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6bda373-d3b3-416e-bc4e-13a99b192c75%2F6ad5dcb0-341a-4a59-a849-ee45e31ba9c9%2Fota8bo_processed.png&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Purposes
**Joint Probability Density Function:**
\[ f_{X,Y}(x,y) = \frac{6}{5} (x + y^2), \quad 0 \le x \le 1, \, 0 \le y \le 1 \]
**Questions:**
1. **a)** What is the marginal probability density function (pdf) of \( f_Y(y) \)?
2. **b)** What is the conditional pdf of \( f_{X|Y}(x|y = 0.5) \)?
**Q3. Exponential Distribution Properties:**
The Exponential Distribution has a *memoryless* property. Intuitively, this means that the probability of a customer service answering your call (assuming waiting time is exponential) in the next 10 minutes is the same, regardless of whether you have waited an hour on the line or just picked up the phone. Formally, if \( X \sim \text{exponential}(\lambda) \), then \( f(x) = \lambda \exp(-\lambda x) \). Let \( t \) and \( s \) be two positive numbers. Use the definition of conditional probability to show:
\[ P(X > t + s \mid X > t) = P(X > s) \]
*Hint:* Find the cumulative distribution function (cdf) of \( X \) first, and note that \( P(X > t + s \cap X > t) = P(X > t + s) \).
**Q4. Fair Die Game:**
Roll a fair die (uniform distribution over 1, 2, 3, 4, 5, 6) repeatedly. You and Peter are betting on the number shown on each roll.
- If the number is 4 or less, you win $1; otherwise, you pay Peter $2.5.
- **a)** What is the expected value of the payoff for you?
- **b)** What is the variance of your payoff?
**Q5. Uniform Distribution:**
Let \( X \) be uniformly distributed (continuous) on the interval \([1,2]\). Find \( E(1/X) \).
**Q6. Exponential Distribution for Light Bulbs:**
Assume the lifetime of light bulbs follows an exponential distribution with pdf:
\[ f(x) = \lambda \exp(-\lambda x) \
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