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 x in a manner that and y have the same mean. In particular, we can let 9 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 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 : x a₁ a₂ a3 04 f(x) Assume that a₁ < a < 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 x in a manner that and y have the same mean. In particular, we can let 9 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 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 : x a₁ a₂ a3 04 f(x) Assume that a₁ < a < a
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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Transcribed Image Text:In general, consider a discrete random variable with probability mass
function f which gives the outcome a with probability P₁, 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 x 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 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 *:
x
a1 az a3 04
ƒ(x)
Assume that a₁ < A₂ < az < A4 and:
Qg =
a1 + as
2
a3 =
a₂ + as
2
Construct a random variable y with probability mass function 9:
y
51 | 52 | 53 | 54
g(y)H
=
where by: 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 x but shifts
the probability mass from the center of f towards the tails.
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