Binominal Distributions Worksheet With Answer Key - Helm 2008 Section 372 Page 11
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2. Expectation and variance of the binomial distribution
For a binomial distribution X
B(n, p), the mean and variance, as we shall see, have a simple form.
While we will not prove the formulae in general terms - the algebra can be rather tedious - we will
illustrate the results for cases involving small values of n.
The case n n n = = = 2 2 2
Essentially, we have a random variable X which follows a binomial distribution X
B(2, p) so that
2
the values taken by X (and X
- needed to calculate the variance) are shown in the following table:
2
2
x
x
P(X = x)
xP(X = x)
x
P(X = x)
2
0
0
q
0
0
1
1
2qp
2qp
2qp
2
2
2
2
4
p
2p
4p
We can now calculate the mean of this distribution:
2
E(X) =
xP(X = x) = 0 + 2qp + 2p
= 2p(q + p) = 2p
since q + p = 1
Similarly, the variance V (X) is given by
2
2
2
2
V (X) = E(X
)
[E(X)]
= 0 + 2qp + 4p
(2p)
= 2qp
Task
Calculate the mean and variance of a random variable X which follows a binomial
distribution X
B(3, p).
Your solution
27
HELM (2008):
Section 37.2: The Binomial Distribution
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