Probability Generating Functions Flashcards
If a discrete random variable, X, has a probability mass function P(X=x)…
- The the probability generating function of X is given by Gx(t) = Σ P(X=x)t*
- Where * is to the power of x, and t is a dummy variable
For any probability generating function Gx(1)
= 1
For the probability table below what is the probability generating function
For the probability table below what is the probability generating function
P(X=x) for binomial distribution X~B(n,p)
Probability generating function for binomial distribution, X~B(n,p)
Distribution for Poisson, X~Po(λ)
Probability generating function for the Poisson distribution
Geometric distribution formula
Probability generating function for the geometric distribution
Negative binomial formula, X~Negative B(r,p)
Probability generating function for the Negative binomial, X~Negative B(r,p)
For a discrete random variable with probability generating function Gx(t),
E(X) =
For a discrete random variable with probability generating function Gx(t),
Var(X) =
If X and Y are two indépendant random variables with probability generating function Gx(t) and Gy(t),
The probability generating function of Z = X + Y is given by
Gz(t) = Gx(t) x GY(t)
If the discrete random variable X has probability generating function Gx(t),
Then the probability generating function of the discrete random variable Y = aX + b, where a and b are positive integers is given by
