'You are a spy who is about to be exposed' distribution

Question 1

how to find discrete variance

Answer 1

<x^2> - <x>^2
or
sum [ (x - <x>)^2 * P(x) ]
(expectation value of squared difference from <x>)

Question 2

how to find discrete expectation value

Answer 2

sum [ x * P(x) ]

Question 3

how to find expectation value of f(x) for prob mass function P(x)

Answer 3

sum [ f(x) * P(x) ]

Question 4

bernoulli distribution

Answer 4

• sample space omega = {0, 1}
• P(x|p) = p^x * (1 - p)^(1 - x)
• <x> = p
• var(x) = p(1 - p)

Question 5

binomial distribution

Answer 5

• P(k|N, p) = NCk * p^k * (1 - p)^(N - k)
• sample space omega = {0, 1, … , N - 1, N}
• <k> = Np
• var(k) = Np(1 - p)
• <k> and var(k) are just N times bernoulli

Question 6

poisson distribution

Answer 6

• limit of binomial for N->inf, p->0, lambda = Np
• P(k|lambda) = lambda^k * exp(- lambda) / k!
• <k> = lambda
• var(k) = lambda
• <k> and var(k) same as binomial but applying p->0

Question 7

nth central moment

Answer 7

mu_n = <(x - <x>)^n>
(var(x) is 2nd central moment)