Download Mean of a Continuous Random Variable

Continuous Random Variables
Page 1
Mean of a Continuous Random Variable
The goal of this activity is to understand the mean or expected value of a continuous random
variable.
0.30
The figure on the right
represents a relative
frequency histogram of n
observation organized in m
intervals of width ∆x and
defined by the function h(x).
In addition, h(x)
approximates the probability
density function f(x) of a
continuous random variable
from a to b.
y = f(x)
0.25
0.20
0.15
h(x4)
0.10
0.05
h(x3)
h(x1)
0.00
0
a =1 x1
h(x2)
x22
h(xm)
x33
x44
5
6 . 7. .
8
9
xm10
=b
11
Recall that the mean of a discrete random variable X is given by µ X = ∑ [x ⋅ P(x )], where x is
the value of the random variable and P(x) is the probability of observing the random variable x.
m
Thus, the mean of the relative frequency histogram is given by µ X = ∑ [ xi ⋅ h ( xi )∆x ] .
i =1
Let n grow without bound, which subsequently decreases interval ∆x without bound. Then, the
function h(x) approaches f(x) and we get
µ X = lim ∑ xf ( xi )∆x
∆x →0
This results in the following definition.
The Mean of a Continuous Random Variable
The mean (or expected value) of a continuous random variable X with a probability density
function f(x) is given by the formula
µX = E( X ) =
∞
∫ xf ( x )dx
−∞
Robert A. Powers
University of Northern Colorado
Continuous Random Variables
Page 2
1. Find the mean of uniform probability distribution from 0 to 1 given by the probability density
function U(0,1), where
0, if x < 0

U (0,1) = 1, if 0 ≤ x < 1
 0, if x ≥ 1

2. Let the random variable X be the lengths of time in minutes between calls to 911 in a small
city that were reported in the newspaper over a two day period. Suppose that a reasonable
probability model for X is given by the probability density function (p.d.f.) below.
1 − x / 20
f ( x) =
e
,0≤x<∞
20
Find the expected time between calls.
3. Let Y be a continuous random variable with p.d.f. g(y) = 2y, 0 < y < 1.
Find the mean of Y.
4. Let Z be a continuous random variable with p.d.f h ( z ) =
1
π ( x 2 + 1)
for z ∈ R.
Find the mean of Z.
Challenge Problem
A triangle probability distribution function on the interval [a,b] is
shown in the figure on the right. Find the mean of the triangle
distribution.
Robert A. Powers
a
b
University of Northern Colorado
Continuous Random Variables
Page 3
Standard Deviation of a Continuous Random Variable
The goal of this activity is to understand the variance and standard deviation of a continuous
random variable.
0.30
The figure on the right
represents a relative
frequency histogram of n
observation organized in m
intervals of width ∆x and
defined by the function h(x).
In addition, h(x)
approximates the probability
density function f(x) of a
continuous random variable
from a to b.
y = f(x)
0.25
0.20
0.15
h(x4)
0.10
0.05
h(x3)
h(x1)
0.00
0
a =1 x1
h(x2)
x22
h(xm)
x33
x44
5
6 . 7. .
8
9
xm10
=b
11
Recall that the variance of a discrete random variable X is given by σ X2 = ∑ [( x − µ X ) 2 ⋅ P( x )] ,
where x is the value of the random variable, µX is the mean of the random variable, and P(x) is
the probability of observing the random variable x.
m
Thus, the mean of the relative frequency histogram is given by σ X2 = ∑ [( xi − µ X ) 2 ⋅ h ( xi )∆x ] .
i =1
Let n grow without bound, which subsequently decreases interval ∆x without bound. Then, the
function h(x) approaches f(x) and we get
σ X2 = lim ∑ ( xi − µ X ) 2 f ( xi ) ∆x
∆x → 0
This results in the following definition.
The Variance and Standard Deviation of a Continuous Random Variable
The variance of a continuous random variable X with a probability density function f(x) is
given by the formula
∞
σ = Var ( X ) = E [( X − µ X ) ] = ∫ ( x − µ X ) 2 f ( x )dx
2
X
2
−∞
The standard deviation of X is
σ = Var ( X )
Robert A. Powers
University of Northern Colorado
Continuous Random Variables
Page 4
Note: an equivalent form of the variance of X is given by
σ =
2
X
∞
∫x
2
f ( x )dx − µ X2
−∞
1. Find the standard deviation of uniform probability distribution from 0 to 1 given by the
probability density function U(0,1), where
0, if x < 0

U (0,1) = 1, if 0 ≤ x < 1
 0, if x ≥ 1

2. Let the random variable X be the lengths of time in minutes between calls to 911 in a small
city that were reported in the newspaper over a two day period. Suppose that a reasonable
probability model for X is given by the probability density function (p.d.f.) below.
1 − x / 20
f ( x) =
e
,0≤x<∞
20
Find the standard deviation of the time intervals between calls.
3. Let Y be a continuous random variable with p.d.f. g(y) = 2y, 0 < y < 1.
Find the standard deviation of Y.
4. Let Z be a continuous random variable with p.d.f h ( z ) =
1
π ( x 2 + 1)
for z ∈ R.
Find the standard deviation of Z.
Challenge Problem
Find the standard deviation of the triangle distribution on the interval [a,b].
Robert A. Powers
University of Northern Colorado