Course Notes - Department of Mathematics and Statistics

Mean of Binary DistributionThe mean is colliquially defined as what is the average outcome ofan event.Mean = µ Y = 1 × π + 0 × (1 − π) = πVariance of Binary DistributionVariance(σY 2 ) = (1 − π)2 × π + (0 − π) 2 × (1 − π)= π (1 − π) 2 + π 2 (1 − π)= π (1 − π) (1 − π + π)= π (1 − π)Distribution of the Binomial distribution• Suppose we take a sample of size n from the underlying populationand look at the distribution of the number of successes.• Total number of successes :X = Y 1 + Y 2 + Y 3 + . . . + Y n• Where all the Y i ’s are independent of each other.• This is the Binomial distribution.Mean and Variance of the Binomial distributionCombining random variables gives the mean of X as:The variance of X is given as:µ X = π Y1 + π Y2 + π Y3 + . . . + π Ynσ 2 X = σ2 Y 1+ σ 2 Y 2+ σ 2 Y 3+ . . . + σ 2 Y n61