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Lecture 17
Zhihua (Sophia) Su
University of Florida
Feb 17, 2015
STA 4321/5325 Introduction to Probability 1
Agenda
Expected Values of Continuous Random VariablesTchebysheff’s Inequality for Continuous Random VariablesExamples
Reading assignment: Chapter 4: 4.3
STA 4321/5325 Introduction to Probability 2
Expected Values of Continuous RandomVariables
DefinitionIf X is a continuous random variable with density fX , then theexpected value of X is defined as
E(X) =
ˆ ∞−∞
xfX(x)dx.
(Assuming that´∞−∞ | x | fX(x)dx <∞.)
Notice that it parallels with discrete random variables. The sumis replaced by an integral, and the probability mass function pXis replaced by the probability density function fX .
STA 4321/5325 Introduction to Probability 3
Expected Values of Continuous RandomVariables
ResultIf X is a continuous random variable with density fX , then forany function g: R→ R,
E[g(X)] =
ˆ ∞−∞
g(x)fX(x)dx.
(Assuming that´∞−∞ | g(x) | fX(x)dx <∞.)
STA 4321/5325 Introduction to Probability 4
Expected Values of Continuous RandomVariables
The variance of a continuous random variable is defined in asimilar fashion as a discrete random variable.
DefinitionIf X is a continuous random variable with probability densityfunction fX , then the variance of X is given by
V (X) = E[(X − E(X))2]
=
ˆ ∞−∞
(x− E(X))2fX(x)dx.
STA 4321/5325 Introduction to Probability 5
Expected Values of Continuous RandomVariablesLet us look at an example to understand these definitions andresults.
Example: For a given teller in a bank, let X denote theproportion of time, out of a 40-hour work week, that he isdirectly serving the customers. Suppose that X has aprobability density function given by
fX(x) =
{3x2 if 0 ≤ x ≤ 1,
0 otherwise.
(a) Find the mean proportion of time during a 40-hour workweek the teller directly serves customers.
STA 4321/5325 Introduction to Probability 6
Expected Values of Continuous RandomVariables
Example (cont’d)(b) Find the variance of the proportion of time during a 40-hourwork week the teller directly serves customers.
STA 4321/5325 Introduction to Probability 7
Expected Values of Continuous RandomVariables
There is another way to determine expectations of non-negativecontinuous random variables directly from their distributionfunctions.
ResultIf X is a non-negative continuous random variable, then
E(X) =
ˆ ∞0
P (X > x)dx =
ˆ ∞0
[1− FX(x)]dx.
STA 4321/5325 Introduction to Probability 8
Tchebysheff’s Inequality for ContinuousRandom Variables
The Tchebysheff’s theorem holds for continuous randomvariables in the same way as for discrete random variables.
ResultIf X is a continuous random variable with E(X) = µX andSD(X) = σX , then for any k ≥ 0,
P (| X − µX |< kσX) ≥ 1− 1
k2.
STA 4321/5325 Introduction to Probability 9
Examples
Example: The distribution function of the random variable X,the time (in years) from the time a machine is serviced until itbreaks down, is as follows:
FX(x) =
{1− e−4x if x > 0,
0 otherwise.
(a) Find E(X), V (X).
STA 4321/5325 Introduction to Probability 10
Examples
(b) Find an interval such that the probability that X lies in theinterval is at least 75%.
STA 4321/5325 Introduction to Probability 11
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