Binomial distribution

What are the 4 conditions to model x as distributed binomialy

1) fixed number of trials

2) two possible outcomes (success or failure)

3) there is a fixed probability of success

4) the trials are independent of eachother

x ~ B(n, p)

P(x = 4) =

ⁿc₄ x p⁴ x (1-p)ⁿ⁻⁴

when does binomial distribution have vertical symetry

when the probability = 0.5

y ~ B(n, p)

P(y = 3) = 0.45

x ~ B(n, 1-p)

what does P(x = n-3)

P(x = n-3) = 0.45

because B(n,p) and B(n,1-p) have reflective vertical symmetry

binomial pd vs cd

pd is when P(X = x)

cd is when P(X ≤ x)

P(X < x) =

P(X ≤ x -1)

P(X ≥ x)

1 - P(X ≤ x -1)

P(X > x)

1 - P(X ≤ x)

P(y ≤ X < z)

P(X ≤ z-1) - P(X ≤ y-1)

how to test a binomial distribution

mean = np

variance = np(1-p)

in formu