1. Probability theory 2010 Outline  The need for transforms  Probability-generating function  Moment-generating function  Characteristic function  Applications of transforms to branching processes

2. Probability theory 2011 Definition of transform  In probability theory, a transform is function that uniquely determines the probability distribution of a random variable An example:.

3. Probability theory 2011 Using transforms to determine the distribution of a sum of random variables

4. Probability theory 2011 The probability generating function Let X be an integer-valued nonnegative random variable. The (probability) generating function of X is  Defined at least for | t | < 1  Determines the probability function of X uniquely  Adding independent variables corresponds to multiplying their generating functions Example 1: X  Be(p) Example 2: X  Bin(n;p) Example 3: X  Po(λ) Addition theorems for binomial and Poisson distributions

5. Probability theory 2011 The moment generating function Let X be a random variable. The moment generating function of X is provided that this expectation is finite for | t | 0  Determines the probability function of X uniquely  Adding independent variables corresponds to multiplying their moment generating functions

6. Probability theory 2011 The moment generating function and the Laplace transform Let X be a non-negative random variable. Then

7. Probability theory 2011 The moment generating function - examples The moment generating function of X is Example 1: X  Be(p) Example 2: X  Exp(a) Example 3: X   (2;a)

8. Probability theory 2011 The moment generating function - calculation of moments

9. Probability theory 2011 The moment generating function - uniqueness

10. Probability theory 2011 Normal approximation of a binomial distribution Let X 1, X 2, …. be independent and Be(p) and let Then But

11. Probability theory 2011 The characteristic function Let X be a random variable. The characteristic function of X is  Exists for all random variables  Determines the probability function of X uniquely  Adding independent variables corresponds to multiplying their characteristic functions

12. Probability theory 2011 Comparison of the characteristic function and the moment generating function Example 1: Exp(λ) Example 2: Po(λ) Example 3: N(  ;  ) Is it always true that.

13. Probability theory 2011 The characteristic function - uniqueness For discrete distributions we have For continuous distributions with we have.

14. Probability theory 2011 The characteristic function - calculation of moments If the k:th moment exists we have.

15. Probability theory 2011 Using a normal distribution to approximate a Poisson distribution Let X  Po(m) and set Then.

16. Probability theory 2011 Using a Poisson distribution to approximate a Binomial distribution Let X  Bin(n ; p) Then If p = 1/n we get.

17. Probability theory 2011 Sums of a stochastic number of stochastic variables Condition on N and determine: Probability generating function Moment generating function Characteristic function

18. Probability theory 2011 Branching processes  Suppose that each individual produces j new offspring with probability p j, j ≥ 0, independently of the number produced by any other individual.  Let X n denote the size of the n th generation  Then where Z i represents the number of offspring of the i th individual of the ( n - 1 ) st generation. generation

19. Probability theory 2011 Generating function of a branching processes Let X n denote the number of individuals in the n:th generation of a population, and assume that where Y k, k = 1, 2, … are i.i.d. and independent of X n Then Example:

20. Probability theory 2011 Branching processes - mean and variance of generation size  Consider a branching process for which X 0 = 1, and  and  respectively depict the expectation and standard deviation of the offspring distribution.  Then.

21. Probability theory 2011 Branching processes - extinction probability  Let  0 = P(population dies out ) and assume that X 0 = 1  Then where g is the probability generating function of the offspring distribution

22. Probability theory 2011 Exercises: Chapter III 3.1, 3.6, 3.9, 3.15, 3.26, 3.35, 3.45