1. Mean & Variance for the Binomial Distribution
We can calculate the probability distribution for a binomial situation If we look at tossing a coin 4 times and finding the number of heads. We have n=4 π = 0.5 so X~B(4,0.5) We therefore have the probability distribution values from tables μ=E(X)=0x x x x x = 2 =4x0.5 =nπ x 1 2 3 4
P(X=x)
0.0625
0.2500
0.3750

2. σ2 = E(X2)- μ2 =(02x x x x x0.0625)-4 = 5-4 = 1 =4x0.5x0.5 = n π(1- π) We could repeat with other cases to show that μ= n π and σ2 = n π(1- π) σ = √nπ(1- π) eg A machine produces goods which have a 1% chance of being faulty. Find the mean number of faulty goods per batch of 1000, and the standard deviation in the number of faulty goods. X~B(1000,0.01) μ= n π = 1000x0.01 =10
σ = √nπ(1- π)
= √1000x0.01x0.99
= √9.9 =