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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
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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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.
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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