If the moment generating function of X is given by M(t) = exp(500t + 5000t²). a) Find the mean and standard deviation of X. b) Find P(X>650). c) Find P(X <710|X > 650). d) Find P(27060 ≤ (X - 500)² ≤ 50240). e) Find the mean of (X - 500)², i.e. E((X - 500)²).
If the moment generating function of X is given by M(t) = exp(500t + 5000t²). a) Find the mean and standard deviation of X. b) Find P(X>650). c) Find P(X <710|X > 650). d) Find P(27060 ≤ (X - 500)² ≤ 50240). e) Find the mean of (X - 500)², i.e. E((X - 500)²).
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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part d, e
![**Problem 7: Analysis of a Random Variable Using Moment Generating Functions**
Given: The moment generating function (MGF) of a random variable \( X \) is defined as \( M(t) = \exp(500t + 5000t^2) \).
**Tasks:**
a) Determine the mean and standard deviation of \( X \).
b) Calculate the probability \( P(X > 650) \).
c) Evaluate the conditional probability \( P(X < 710 | X > 650) \).
d) Find the probability \( P(27060 \leq (X - 500)^2 \leq 50240) \).
e) Compute the expected value of \( (X - 500)^2 \), denoted as \( E((X - 500)^2) \).
---
**Explanations:**
- **Moment Generating Function (MGF):** The MGF is a function that generates the moments of a random variable, providing information about its mean and variance.
- **Mean:** The first derivative of the MGF with respect to \( t \), evaluated at \( t = 0 \), gives the mean of the random variable.
- **Variance:** The second derivative of the MGF at \( t=0 \) minus the square of the mean gives the variance. The square root of the variance gives the standard deviation.
- **Conditional Probability:** Represents the probability of an event occurring given that another event has already occurred.
- **Expected Value of a Function:** Denoted by \( E[g(X)] \), it involves finding the mean of a transformed random variable.
This analysis involves finding probabilities and expected values using properties of the moment generating function, crucial in understanding the distribution and behavior of random variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12050043-4d02-4d31-b9b4-7dfbc30a9c5d%2F67ad1c79-e251-4f28-9dab-0e94c53c15a0%2Fa1b5hu_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7: Analysis of a Random Variable Using Moment Generating Functions**
Given: The moment generating function (MGF) of a random variable \( X \) is defined as \( M(t) = \exp(500t + 5000t^2) \).
**Tasks:**
a) Determine the mean and standard deviation of \( X \).
b) Calculate the probability \( P(X > 650) \).
c) Evaluate the conditional probability \( P(X < 710 | X > 650) \).
d) Find the probability \( P(27060 \leq (X - 500)^2 \leq 50240) \).
e) Compute the expected value of \( (X - 500)^2 \), denoted as \( E((X - 500)^2) \).
---
**Explanations:**
- **Moment Generating Function (MGF):** The MGF is a function that generates the moments of a random variable, providing information about its mean and variance.
- **Mean:** The first derivative of the MGF with respect to \( t \), evaluated at \( t = 0 \), gives the mean of the random variable.
- **Variance:** The second derivative of the MGF at \( t=0 \) minus the square of the mean gives the variance. The square root of the variance gives the standard deviation.
- **Conditional Probability:** Represents the probability of an event occurring given that another event has already occurred.
- **Expected Value of a Function:** Denoted by \( E[g(X)] \), it involves finding the mean of a transformed random variable.
This analysis involves finding probabilities and expected values using properties of the moment generating function, crucial in understanding the distribution and behavior of random variables.
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