"A first Course in probability"-chaper4-discrete random variable-two-item distribution

Two items distributed:

Based on the most basic discrete random variable--Bernoulli random variable x, we perform n repetition experiments, the probability distribution result is so-called two-item distribution.

Specifically, the probability of a successful experiment is p, and now we do n at this time Yang, set random variable x to indicate the number of successful times of n experiments, then the following distribution column is established.

Regarding its expectation, derivation process and geometric distribution, the expected derivation of hypergeometric distribution is homogeneous, first exit x^k expression, then according to the two-item identity relationship, seek self-similarity to set up a recursive relationship, and then get the final expectation, because the detailed derivation process has been given before, This is no longer to repeat the interest of the reader can be manually deduced.

You will end up with a formula like this:

It is not difficult to see when K=1, E[X]=NP.

On monotonicity of probability value of two-item distribution here is a proposition: for the two random variables that satisfy the parameter (n,p), when K obtains [0,n], p{x=k} increments first, then decrements, and when k = (n+1) p gets the maximum value.

"A first Course in probability"-chaper4-discrete random variable-two-item distribution