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In this problem we will explore the properties of the Probability Mass Function (PMF). Let X be a discrete random variable, and px (r) its PMF. (a) Recall that px (x): R→ [0, 1] (the PMF is a function from the real numbers to [0,1]), and that R is an uncountable set. Why then, in words, can we still use the PMF to characterize a discrete random variable? (b) Formalize your answer in part (a) mathematically by proving the statement: {ER | px(x) 0} is a countable set. (c) In words, explain why necessarily E, Px(x) = 1. (d) Prove the statement in part (c) mathematically, using the fact that P() = 1.

In this problem we will explore the properties of the Probability Mass Function (PMF). Let X be a discrete random variable, and px (r) its PMF. (a) Recall that px (x): R→ [0, 1] (the PMF is a function from the real numbers to [0,1]), and that R is an uncountable set. Why then, in words, can we still use the PMF to characterize a discrete random variable? (b) Formalize your answer in part (a) mathematically by proving the statement: {ER | px(x) 0} is a countable set. (c) In words, explain why necessarily E, Px(x) = 1. (d) Prove the statement in part (c) mathematically, using the fact that P() = 1.

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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In this problem we will explore the properties of the Probability Mass Function (PMF). Let X be a discrete random variable, and px(x) its PMF. (a) Recall that px (x): R→ [0, 1] (the PMF is a function from the real numbers to [0,1]), and that R is an uncountable set. Why then, in words, can we still use the PMF to characterize a discrete random variable? (b) Formalize your answer in part (a) mathematically by proving the statement: { ER | px(x) 0} is a countable set. (c) In words, explain why necessarily Σ, Px(x) = 1. (d) Prove the statement in part (c) mathematically, using the fact that P() = 1.
Transcribed Image Text:In this problem we will explore the properties of the Probability Mass Function (PMF). Let X be a discrete random variable, and px(x) its PMF. (a) Recall that px (x): R→ [0, 1] (the PMF is a function from the real numbers to [0,1]), and that R is an uncountable set. Why then, in words, can we still use the PMF to characterize a discrete random variable? (b) Formalize your answer in part (a) mathematically by proving the statement: { ER | px(x) 0} is a countable set. (c) In words, explain why necessarily Σ, Px(x) = 1. (d) Prove the statement in part (c) mathematically, using the fact that P() = 1.
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