Simple stochastic models 1

Random variation 1 Genetic, physiological Environmental Measurement error Random sampling

Random variation 2 Random variability as nuisance Random variability of primary interest

Terminology Trial Sample space Event Probability Random variable

Discrete random variable X Takes values: x0, x1, x2, …. with corresponding probabilities P(X=xk)=pk p0, p1, p2, … p0 + p1 + p2 + … = 1

Discrete - integer random variable X Takes values: 0, 1, 2, …. with corresponding probabilities: P(X=k)=pk p0, p1, p2, … p0 + p1 + p2 + … = 1

Mean and variance of discrete random variable

Binomial Distribution (Bernoulli trials) n – number of trials p – probability of success P(k successes in n trials) = nCk pk (1-p)n-k

Binomial distribution X ~ binomial(n,p) E(X) = n p V(X) = n p (1 – p )

n = 10, p = 0.5 0.3 0.25 P(X=k) 0.2 0.15 0.1 0.05 1 2 3 4 5 6 7 8 9 10 k

Poisson distribution X ~ poisson() k – number of events E(X)= , V(X)= 

 = 5 0.18 0.16 0.14 P(X=k) 0.12 0.1 0.08 0.06 0.04 0.02 2 4 6 8 10 12 14 16 18 20 k

When number of Bernoulli trials n is large, and probability of success p is small, the distribution of number of successes becomes Poisson.

Examples 30 % of women in Germany are smokers. We take a random (representative) sample of 20 women. The distribution of number of smokers among them is X ~ binomial(20,0.3)

Probability that an accident leading to injury happens in a factory is p=0.0001 per day. Number of accidents X over a ten year period (n=3650) is Poisson, X~Poisson(), with  = np = 0.365

Generating function of discrete - integer random variables X: 0, 1, 2, …. p0, p1, p2, … Generating function: P(s) = p0 + s p1 + s2 p2 + …=

Properties P(1) = 1 P’(1) = E(X) P’’(1) = E[X(X-1)]

Binomial and Poisson distrib. Binomial: X ~ binomial(n,p), q=1-p Poisson: X ~ Poisson( )

Continuity theorem for generating functions

Two – dimensional discrete – integer random variables Joint probability distribution PXY(X=i, Y=k) = pik Two – dimensional generating function: Marginal distributions: PX(s)=PXY(s,1) PY(s)=PXY(1,s)

Independent discrete – integer random variables X: PX(X=i) = pi Y: PY(Y=k) = pk PXY(X=i, Y=k) = pi pk

Sum PXY(X=i, Y=k) = pik Z = X + Y

Expectation and variance of sums of independent random variables E(X+Y) = E(X) + E(Y) X, Y - independent V(X+Y) = V(X) + V(Y)

Sum Z = X + Y PZ(s) = PXY(s,s) X,Y – independent: PZ(s) = PX(s) PY(s)

Example X1 ~ binomial(1,p) (one Bernoulli trial) Xn ~ binomial(n,p) PX1(s)=q+sp PXn(s)=(q+sp)n

Example X ~ Poisson(), Y~Poisson() Z=X+Y PZ(s) = PX(s) PY(s) Z ~ Poisson(+)

Can we generalize generating function method to non integer discrete r Can we generalize generating function method to non integer discrete r.v. ? x0, x1, x2, …. P(X=xk)=pk: p0, p1, p2, …