Lecture 8

Zhihua (Sophia) Su

University of Florida

Jan 26, 2015

STA 4321/5325 Introduction to Probability 1

Agenda

Expected ValuesVarianceProperties of Expected Values and Variance

Reading assignment: Chapter 3: 3.3, 3.11

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Expected Values

Suppose we are interested in a random variable X arising out ofa random experiment. Based on our understanding of therandom experiment, we have a probability model. Often, wewant to summarize our understanding of the random variable inone number, “the expected value” of the random variable.

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Expected Values

DefinitionThe expected value of a discrete random variable X withprobability mass function pX is given by

E(X) =∑x∈X

xpX(x) =∑x∈X

xP (X = x).

It is also understood as our estimate of the “average” value thatthe random variable will take.

Note: The expected value of a discrete random variable isdefined only if

∑x∈X | x | P (X = x) <∞.

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Expected Values

Example: Consider the following game. We toss a six-faced die2 times. If the sum of the two values is 3 or lower, we have topay 10 dollars. If the sum of the two values is 4, 5 or 6, we pay4 dollars. If the sum of the two values is 7, 8 or 9, we gain 4dollars. If the sum of the two values is 10, 11 or 12, we gain 10dollars. What are the expected winning?

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Expected Values

ResultIf X is a discrete random variable with probability distributionpX(x), and if g: R→ R is a real-valued function, then

E(g(X)) =∑x∈X

g(x)pX(x).

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Expected Values

Example: In the last example, if g(x) = x2, find E(g(W )).

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Variance

DefinitionThe variance of a random variable X with expected value µ isgiven by

V (X) = E(X − µ)2 =∑x∈X

(x− µ)2pX(x).

Note that the variance V (X) of a random variable is the averagesquared distance between the values of X and the expectedvalue µ.

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Variance

DefinitionThe standard deviation of a random variable X is the squareroot of the variance and is given by

SD(X) =√

E(X − µ)2.

The standard deviation is also a measure of the variability of arandom variable, but it maintains the original unites of measure.It can be thought of as the size of a typical deviation betweenan observed outcome and the expected value.

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Variance

Example: If W is the winnings in the game discussed in theprevious lecture, find the variance and standard deviation of W .

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Properties of Expected Values and Variance

Result 1V (aX + b) = a2V (X).

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Properties of Expected Values and Variance

Result 2V (X) = E(X2)− (E(X))2.

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Properties of Expected Values and Variance

Result 3 (Tchebysheff’s Theorem)

Let X be a random variable with mean µ and variance σ2.Then for any positive k,

P (| X − µ |< kσ) ≥ 1− 1

k2.

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