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When does the sum of independent variables follow a binomial distribution?
The sum of independent variables each following binomial distributions B ( N i, p i) is also binomial if all p i = p are equal (in this case the sum follows B ( ∑ i N i, p). If the p i are distinct, the sum follows the more general Poisson-Binomial distribution.
How are independent and identically distributed random variables different?
Then “independent and identically distributed” implies that an element in the sequence is independent of the random variables that came before it. In this way, an i.i.d. sequence is different from a Markov sequence, where the probability distribution for the n th random variable is a function of the previous random variable in the sequence
Which is the sum of N I Bernoulli variables?
Since a binomial B ( N i, p i) is the sum of N i Bernoulli variables with success probability p i, it follows that the sum of binomials considered above is equal to the Poisson-Binomial, with parameters p 1, p 1, … (repeated N 1 times), p 2, p 2, … (repeated N 2 times)., being the sum of ∑ i N i Bernoulli variables.
What is the Poisson binomial with parameters q k?
The Poisson-Binomial with parameters q k is the distribution of the sum of Bernoulli variables with success probabilities q k.
Is the variance of the binomial distribution equal to P?
varianceis equal to p(1-p). By the addition properties for independent random variables, the mean and variance of the binomial distribution are equal to the sum of the means and variances of the nindependent Zvariables, so
How to combine two binomial random variables in R?
This answer provides an R implementation of the explicit formula from the paper linked in the accepted answer ( The Distribution of a Sum of Binomial Random Variables by Ken Butler and Michael Stephens). (This code can in fact be used to combine any two independent probability distributions):
Is the distribution function of a sum of independent variables symmetric?
The distribution function of a sum of independent variables isDifferentiating both sides and using the fact that the density function is the derivative of the distribution function, we obtainThe second formula is symmetric to the first. The two integrals above are called convolutions (of two probability density functions).