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Transcript
The Mean and Standard Deviation of a Finite Distribution
I. The idea
We want to have a measure of the center and the spread of a finite random
variable whose probabilities are given by a probability mass function. Since
we can think of the distribution as giving the probabilities for an enormous
number of repetitions of the experiment which provides the random variable,
this is equivalent to finding the population mean and standard deviation.
The symbols for these are µ and σ, respectively, and if there are more than
one random variable wandering about, subscripts are used, i.e. µX and σY .
Note that these are not the X n and s which come from samples.
II. The mean
The formal definition is
n
X
xi · P (X = xi ).
i=1
The most convenient way to mechanize this is in a table which displays the
probability model and organizes the calculation of the sum.
i
xi
1
2
2
4
3
6
Sum
P (X = xi ) xi · P (X = xi )
.40
.8
.30
1.2
.30
1.8
1.00
3.8
Thus in the case of this random variable, µ = 0.8 + 1.2 + 1.8 = 3.8. This is
the long-term average outcome if I run the experiment very many times.
III. Variance and Standard Deviation
As with samples, there are two measures of spread for a random variable.
The first is called distribution variance VX or VAR(X). The idea is similar to
the calculations from a sample, but just enough different to cause problems.
We have
n
VAR(X) =
X
(xi − µ)2 · P (X = xi )
i=1
and
σ=
q
VAR(X)
We can calculate these things by extending the above table in a fairly obvious
way, and I will use the same example.
i
1
2
3
Sums
xi
2
4
6
P (X = xi ) xi · P (X = xi ) xi − µ
.40
.8
-1.8
.30
1.2
.2
.30
1.8
2.2
1.00
3.8
1
(xi − µ)2
3.24
0.04
4.84
(xi − µ)2 · P (X = xi )
1.2960
0.0120
1.4520
2.7600
Thus
√ the variance of this distribution is 2.7600, and its standard deviation
is 2.7600 = 1.6613.
IV. Binomials
A calculation using the rules of sigma notation yields the following results
for a Binomial Distribution with n repetitions and probabilty of success p:
µ
VAR
σ
where q
=
=
=
=
np
npq
√
npq
1−p
V. Continuous Random Variables
Similar calculations for continuous random variables require calculus, and
are therefore not a part of this course. The results of such calculations will
be supplied as we need them.
2