special class of random variable
X = number of success given N trial with p probability of each success on each trial where each trial are independent with each other.
properties:
- made of independent trials
- each trials can be classified as success or failure
- fixed # of trials
- probability of success on each trial is constant
Binomial distribution
special case of probability mass function from a binomial variable pmf if the probability of success is p, the probability of having exactly k success given n trial is given by:
$$Binom(k|n,p) \triangleq (_kC_n) p^k(1-p)^{n-k}$$
Info
title:Intuition: the probability to get 3 head from 5 trial of a coin with 0.6 probability of having head is
the sum of all probability of having 3 head and 2 coin in every possible combination hence: p(HHHTT) + p(HHTHT) + p(HHTTH) + … + p(TTHHH) with each has probability of = $$p(head)^3 \neg p(head)^2 = 0.6^{3}0.4^{2}$$ and the combination of having 3 Head from 5 trial is denoted as $$_3C_5 = 10$$ so the probability of having exactly 3 head is 0.3456
Mean of Binom
$$E[Binom(k|n,p)] = np$$
Variance of Binom
$$Var[Binom(k|n,p)] = np(1-p)$$