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5. Derive the expected value of the random variable X with distribution Negative Binomial(r, p).

5. Derive the expected value of the random variable X with distribution Negative Binomial(r, p).

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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Show all necessary solutions. 1. A random variable X denotes the number of books bought by a customer in a bookstore. We know that the random variable has the distribution P(0 < X< 1) = E P(1 <X < 2) = , P(0 < X < 3) = , P(X = 3) = P(X > 4). %3D 12 Compute P(X = i) for every i = 1, 2, 3. 2. Let F be given by if t < 0, a if t € [0, 2), 2c +b if t e [2, 3), F(t) = d if t > 3. a. Under which conditions on a, b, c, d is the function F a cumulative distribution function? b. Assume that F is the c.d.f. of an r.v. X. Compute the law of X. 3. Let X ~ Geometric(p) with pe (0, 1). Prove that for every a, b eN we have P(X = a + b|X > a) = P(X = b). 4. Let X be a discrete random variable with the following distribution: P(X = -3) = a, P(X = -2) = , P(X = -1) = b, P(X = 0) = 6, P(X =1) = c, P(X = 2) = 2 a. Find a, b, c. b. Calculate E(3+ 2X), E(3+ 2X), Var(3+ 2X). 5. Derive the expected value of the random variable X with distribution Negative Binomial(r, p).
Transcribed Image Text:Show all necessary solutions. 1. A random variable X denotes the number of books bought by a customer in a bookstore. We know that the random variable has the distribution P(0 < X< 1) = E P(1 <X < 2) = , P(0 < X < 3) = , P(X = 3) = P(X > 4). %3D 12 Compute P(X = i) for every i = 1, 2, 3. 2. Let F be given by if t < 0, a if t € [0, 2), 2c +b if t e [2, 3), F(t) = d if t > 3. a. Under which conditions on a, b, c, d is the function F a cumulative distribution function? b. Assume that F is the c.d.f. of an r.v. X. Compute the law of X. 3. Let X ~ Geometric(p) with pe (0, 1). Prove that for every a, b eN we have P(X = a + b|X > a) = P(X = b). 4. Let X be a discrete random variable with the following distribution: P(X = -3) = a, P(X = -2) = , P(X = -1) = b, P(X = 0) = 6, P(X =1) = c, P(X = 2) = 2 a. Find a, b, c. b. Calculate E(3+ 2X), E(3+ 2X), Var(3+ 2X). 5. Derive the expected value of the random variable X with distribution Negative Binomial(r, p).
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