Discrete Mathematical Structures by Pearson

Combinatorics 5-43

5.13.2 Mean and Variance

Associated with any random variable X, there are two important parameters, that is, mean and

variance. Mean denoted by m gives expectation of X while variance s

2

provides a measure of

dispersion of X.

The expectation of X is

m == =

=

−

=

−

=

−

=

−

=

∑∑

∑

EX xp rcpq

r

n

rn r

pq

nn

ii

i

n

n

r

rnr

r

n

rnr

r

n

() .

.

10

0

1.ppp q

rnr

np

n

rnr

pq

np qp

rnr

r

n

r

n

rn

r

n

.

1

1

1

−−

=

=

−−

−

−−

=

−

−−

=+

∑

∑

1

1

1

1

1

()

==

=

np

np

.1

Hence, mean of binomial distribution b(r: n, p) is np

Now variance

22

== −= −=

−+

==

∑

sm

mm

mEX px px x

ii

i

n

ii i

i

() ()

()

0

22 2

1

2

nn

ii

i

n

ii

i

n

ii

i

n

ii

i

n

px px

px px

∑

∑∑

∑

=− +

=−+=

==

==

2

01

2

2

0

22

1

2

2

mm

mm m.

∑∑

∑

∑

−

=−

=−+−

=

−

=

−

m

2

2

0

22

0

22

1

rcpq np

rr rcpq np

n

r

r

n

rnr

r

n

n

r

rnr

.

[( )]

==−