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11. (a) Find the pmf of X, (b) find the values of E(X), (c) calculate Pr(1 ≤ x ≤ 2) when the moment-generating function of X is given by (a) M (t) = (0.3+0.7e¹)5 (b) M(t) = 0.4et 1-0.6e (c) M(t) = 0.5(e-1) t<-In (0.6).

11. (a) Find the pmf of X, (b) find the values of E(X), (c) calculate Pr(1 ≤ x ≤ 2) when the moment-generating function of X is given by (a) M (t) = (0.3+0.7e¹)5 (b) M(t) = 0.4et 1-0.6e (c) M(t) = 0.5(e-1) t<-In (0.6).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Problem 11: Moment-Generating Function and Probability Calculations** Given the moment-generating function (MGF) \( M(t) \) of a random variable \( X \), perform the following: (a) **Find the Probability Mass Function (pmf) of \( X \)** (b) **Calculate the Expected Value \( E(X) \)** (c) **Calculate the Probability \( \Pr(1 \leq X \leq 2) \)** The moment-generating function \( M(t) \) is given by: (a) \( M(t) = (0.3 + 0.7e^t)^5 \) (b) \( M(t) = \frac{0.4e^t}{1 - 0.6e^t}, \quad t < -\ln(0.6) \) (c) \( M(t) = e^{0.5(e^t - 1)} \) (d) \( M(t) = \sum_{x=1}^5 0.2e^{tx} \) **Instructions:** - Identify which MGF corresponds to standard distributions or known forms. - Use properties of MGFs to find the probability mass function and expectations. - If applicable, apply inverse functions or transformations to find \( \Pr(1 \leq X \leq 2) \).
Transcribed Image Text:**Problem 11: Moment-Generating Function and Probability Calculations** Given the moment-generating function (MGF) \( M(t) \) of a random variable \( X \), perform the following: (a) **Find the Probability Mass Function (pmf) of \( X \)** (b) **Calculate the Expected Value \( E(X) \)** (c) **Calculate the Probability \( \Pr(1 \leq X \leq 2) \)** The moment-generating function \( M(t) \) is given by: (a) \( M(t) = (0.3 + 0.7e^t)^5 \) (b) \( M(t) = \frac{0.4e^t}{1 - 0.6e^t}, \quad t < -\ln(0.6) \) (c) \( M(t) = e^{0.5(e^t - 1)} \) (d) \( M(t) = \sum_{x=1}^5 0.2e^{tx} \) **Instructions:** - Identify which MGF corresponds to standard distributions or known forms. - Use properties of MGFs to find the probability mass function and expectations. - If applicable, apply inverse functions or transformations to find \( \Pr(1 \leq X \leq 2) \).
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