1. Simple stochastic models 1

2. Random variation 1 Genetic, physiological Environmental
Measurement error
Random sampling

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

4. Terminology
Trial
Sample space
Event
Probability
Random variable

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

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

7. Mean and variance of discrete random variable

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

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

10. 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

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

12.  = 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

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

14. 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)

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

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

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

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

19. Continuity theorem for generating functions

20. 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)

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

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

23. 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)

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

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

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

27. 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, …