Related distributions
Related distributions [ edit ] Sums of binomials [ edit ] If X ~ B( n , p ) and Y ~ B( m , p ) are independent binomial variables with the same probability p , then X + Y is again a binomial variable; its distribution is Z=X+Y ~ B( n+m , p ): [26] P ( � = � ) = ∑ � = 0 � [ ( � � ) � � ( 1 − � ) � − � ] [ ( � � − � ) � � − � ( 1 − � ) � − � + � ] = ( � + � � ) � � ( 1 − � ) � + � − � A Binomial distributed random variable X ~ B( n , p ) can be considered as the sum of n Bernoulli distributed random variables. So the sum of two Binomial distributed random variable X ~ B( n , p ) and Y ~ B( m , p ) is equivalent to the sum of n + m Bernoulli distributed random variables, which means Z=X+Y ~ B( n+m , p ). This can also be proven directly using the addition ru...