1. The Binomial Distribution
3.1c
The Binomial Distribution
The Mean and Variance of a Binomial Distribution
X ~ B ( n , p )
The mean of the distribution is the expected value of X E(X) = np .
The random variance of the distribution
Var (X) = npq (q = 1 - p)
The Standard Deviation
√Var (X) = √(npq) (q = 1 - p)

2. c) the mean and variance of Y when Y = 3X + 4
Example
X has a Binomial distribution where the number of trials are 30 and the probability of success is 0.4.
Find
the mean,
the standard deviation,
c) the mean and variance of Y when Y = 3X + 4
X ~ B ( 30 , 0.4 )
a) E(X)
= np
= 30 x 0.4
= 12
b) √Var (X) =
√npq
= √(30 x 0.4 x 0.6)
= √7.2
= 2.68

3. Mathematics Statistics Unit S1 - WJEC
X ~ B ( 30 , 0.4 )
c) E(Y) = E(3x + 4)
= 3E(X) + 4
= 3 x
= 40
Var(Y) = Var(3x + 4)
= 32 Var(X)
= 9 x 7.2
= 64.8
Exercise 4.5c
Mathematics Statistics Unit S1 - WJEC