Home >>STATISTICS, Section 2, binomial distribution 5

The Mean

A random variable X distributed binomially, with trials n and constant probability of success p is described as:

X ~ B(n, p)

By definition the mean is described as,

mean = μ (mu) = E(X) = np

where E(X) is the expectation/expected value of X .

Example

The chance of getting a red sweet from a box of 40 coloured sweets is 1/10 .

How many red sweets would you expect in each box?

Let the random variable of getting a red sweet be X .

Therefore,

X ~ B(40,1/10)

Since the mean/expected value is given by:

μ = E(X) = np

μ = 40 x 1/10 = 4

answer: you would expect 4 red sweets in each box

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Variance

For random variable X distributed binomially, with trials n and constant probability of success p, variance is defined as:

variance = σ² (sigma squared) = Var(X) = np(1 - p)

Sometimes variance is written in terms of the probability of failure q .

Since

p + q = 1

then,

q = (1 - p)

The equation for variance now becomes:

variance = npq

Example

A five sided spinner with numbers 1, 2, 3, 4, 5 on each sector is twirled 20 times and the number of '3' s scored recorded each time.

i) How many times would you expect the '3' to appear?

ii)What is the variance?

i) If X is the random variable distributed binomially,

X ~ B(20,1/5)

μ = E(X) = np

μ =20 x 1/5 = 4

answer: you would expect a '3' to be recorded 4 times

ii) since,

variance = npq

variance = 20 x 1/5 x 4/5 = 80/25 = 3.2

answer: the variance is 3.2

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