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What is the distribution of sum of binomial random variables?
In probability theory and statistics, the sum of independent binomial random variables is itself a binomial random variable if all the component variables share the same success probability. If success probabilities differ, the probability distribution of the sum is not binomial.
What do you mean by Bernoulli distribution?
A Bernoulli distribution is a discrete distribution with only two possible values for the random variable. The distribution has only two possible outcomes and a single trial which is called a Bernoulli trial.
Is there closed expression for sum of Bernoulli variables?
Let x i be independent Bernoulli random variables with success probabilities p i. That is, x i = 1 with probability p i and x i = 0 with probability 1 − p i. Is there a closed expression or an approximate formula for the distribution of the sum ∑ i x i?
How to calculate CDF of sum of independent random variables?
F Z ( z) = P ( Z ≤ z) = P ( X + Y ≤ z) = ∫ R ∫ − ∞ z − x f X, Y ( x, y) d y d x = ∫ R F Y ( z − x) f ( x) d x. where ∗ stands for convolution. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid …
Is there a way to express the cumulative distribution function?
It’s easy to check that the moment generating function ϕ Z ( t) := E [ e i t Z] can be expressed as ϕ Z = ϕ X ⋅ ϕ Y. Is there a way to express the cumulative distribution function F Z ( z) := P ( Z ≤ z) using the cumulative distribution functions of X and Y ? Edit: note I don’t assume that X and Y have a density.
Is the Poisson binomial distribution a generalization?
Yes, in fact, the distribution is known as the Poisson binomial distribution, which is a generalization of the binomial distribution. The distribution’s mean and variance are intuitive and are given by