MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Question

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.
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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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Step 2: Compute P(27060≤(X-500)^2≤50240)
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Step 3: Compute the mean of (X-500)^2
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