Question

The Poisson binomial distribution is the distribution of the sum of independent
Bernoulli random variables that are not necessarily identically distributed. In other words,
the success probability is different for each Bernoulli random variable. Suppose that we have
n Bernoulli trials where the i-th experiment is a success with probability pi. The number of
successes is X.
(a) Calculate the mean and variance of X.
(b) Suppose that n = 4. Calculate P(X = 2).

Answer 1

(a) Since a Poisson binomial distributed variable is a sum of n independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the n Bernoulli distributions:

$$\mu=\displaystyle\sum_{i=1}^np_i$$

$$\sigma^2=\displaystyle\sum_{i=1}^np_i(1-p_i)$$

(2)

$$\mu=p_1+p_2+p_3+p_4$$

$$P(X=2)=\dfrac{e^{-\mu}\cdot\mu^2}{2!}$$