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The Mean, Variance and Standard Deviation of a
Discrete Random Variable
Victor I. Piercey
November 30, 2009
Victor I. Piercey
The Mean, Variance and Standard Deviation of a Discrete Rando
The Mean of a Discrete Random Variable
If you have a discrete random variable X with the following
probability distribution:
Value of X
Probability
x1
p1
x2
p2
···
···
xn
pn
the mean of X is the sum of the products xi pi :
X
µX = x1 p1 + x2 p2 + · · · + xn pn =
xi pi .
This is sometimes called the expected value.
Victor I. Piercey
The Mean, Variance and Standard Deviation of a Discrete Rando
Example
The distribution for X where X is the number of rooms in an
owned house in San Jose, CA is as follows:
Rooms
Probability
Rooms
Probability
1
0.003
6
0.224
2
0.002
7
0.197
3
0.023
8
0.149
4
0.104
9
0.053
5
0.210
10
0.035
Find the mean µX .
Victor I. Piercey
The Mean, Variance and Standard Deviation of a Discrete Rando
Centers and Spreads
The mean measures where the distribution is centered.
The variance and standard distribution, which we will compute
next, measures how the distribution is spread out.
Victor I. Piercey
The Mean, Variance and Standard Deviation of a Discrete Rando
The Variance and Standard Deviation of a Discrete
Random Variable
If you have a discrete random variable X with the following
probability distribution:
Value of X
Probability
x1
p1
x2
p2
···
···
xn
pn
with mean µX , then the variance of X is
σX2 = (x1 −µX )2 p1 +(x2 −µX )2 p2 +· · ·+(xn −µX )2 pn =
X
(xi −µX )2 pi .
The standard deviation σX is the square root of the variance.
Victor I. Piercey
The Mean, Variance and Standard Deviation of a Discrete Rando
Example
The distribution for X where X is the number of rooms in an
owned house in San Jose, CA has mean µX = 6.284 and is as
follows:
Rooms
Probability
Rooms
Probability
1
0.003
6
0.224
2
0.002
7
0.197
3
0.023
8
0.149
4
0.104
9
0.053
5
0.210
10
0.035
Find the variance σX2 and the standard deviation σX .
Victor I. Piercey
The Mean, Variance and Standard Deviation of a Discrete Rando
Assignment
Page 486: 7.23,7.25 and 7.30
Page 491: 7.32, 7.33
Due Wednesday
Victor I. Piercey
The Mean, Variance and Standard Deviation of a Discrete Rando
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