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In general, consider a discrete random variable with probability mass function f which gives the outcome a with probability pi, i = 1,2,..., n. Suppose we want to create a new random variable y with probability mass function g over the outcomes a which is a mean-preserving spread of f. We can create this new random variable y by adding uncorrelated "noise" to ≈ in a manner that and y have the same mean. In particular, we can let g give the outcome b; with probability Pi, i = 1,2, ..., n, where bi is either the amount a for sure or a lottery with expected value equal to a. Then g will be a mean-preserving spread of f. You will now prove this for the following probability mass function for a discrete random variable : a₁ a₂ az as f(x) Assume that a₁ < a

In general, consider a discrete random variable with probability mass function f which gives the outcome a with probability pi, i = 1,2,..., n. Suppose we want to create a new random variable y with probability mass function g over the outcomes a which is a mean-preserving spread of f. We can create this new random variable y by adding uncorrelated "noise" to ≈ in a manner that and y have the same mean. In particular, we can let g give the outcome b; with probability Pi, i = 1,2, ..., n, where bi is either the amount a for sure or a lottery with expected value equal to a. Then g will be a mean-preserving spread of f. You will now prove this for the following probability mass function for a discrete random variable : a₁ a₂ az as f(x) Assume that a₁ < a

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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In general, consider a discrete random variable with probability mass function f which gives the outcome a with probability Pi, i = 1, 2, ...,n. Suppose we want to create a new random variable y with probability mass function g over the outcomes a which is a mean-preserving spread of f. We can create this new random variable y by adding uncorrelated "noise" to in a manner that and y have the same mean. In particular, we can let g give the outcome b; with probability Pi, i = 1,2,..., n, where b; is either the amount a for sure or a lottery with expected value equal to ai. Then g will be a mean-preserving spread of f. You will now prove this for the following probability mass function for a discrete random variable : a1 az a3 04 ƒ(x) Assume that a₁ < A₂ < Az < A4 and: a₂ = Y 9(Y) a1 + a3 a3 = 3 2 a₂ + as 2 Construct a random variable y with probability mass function g: 51 | 52 | 53 | 54 H where bi = a1, b = a4, b₂ is a simple lottery which gives either a1 or as with equal probability, and be is a simple lottery which gives either az or a₂ with equal probability. Show that y has the same mean as " but shifts the probability mass from the center of f towards the tails.
Transcribed Image Text:In general, consider a discrete random variable with probability mass function f which gives the outcome a with probability Pi, i = 1, 2, ...,n. Suppose we want to create a new random variable y with probability mass function g over the outcomes a which is a mean-preserving spread of f. We can create this new random variable y by adding uncorrelated "noise" to in a manner that and y have the same mean. In particular, we can let g give the outcome b; with probability Pi, i = 1,2,..., n, where b; is either the amount a for sure or a lottery with expected value equal to ai. Then g will be a mean-preserving spread of f. You will now prove this for the following probability mass function for a discrete random variable : a1 az a3 04 ƒ(x) Assume that a₁ < A₂ < Az < A4 and: a₂ = Y 9(Y) a1 + a3 a3 = 3 2 a₂ + as 2 Construct a random variable y with probability mass function g: 51 | 52 | 53 | 54 H where bi = a1, b = a4, b₂ is a simple lottery which gives either a1 or as with equal probability, and be is a simple lottery which gives either az or a₂ with equal probability. Show that y has the same mean as " but shifts the probability mass from the center of f towards the tails.
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