1 / 22

Outline

Outline. The need for transforms Probability-generating function Moment-generating function Characteristic function Applications of transforms to branching processes. Definition of transform.

muriel
Télécharger la présentation

Outline

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

  4. 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 Probability theory 2011

  5. 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 | < h, where h > 0 • Determines the probability function of X uniquely • Adding independent variables corresponds to multiplying their moment generating functions Probability theory 2011

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

  7. 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) Probability theory 2011

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

  9. The moment generating function- uniqueness Probability theory 2011

  10. Normal approximation of a binomial distribution Let X1, X2, …. be independent and Be(p) and let Then But Probability theory 2011

  11. 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 Probability theory 2011

  12. 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 . Probability theory 2011

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

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

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

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

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

  18. Branching processes • Suppose that each individual produces j new offspring with probability pj, j≥ 0, independently of the number produced by any other individual. • Let Xn denote the size of the nth generation • Then where Zi represents the number of offspring of the ith individual of the (n - 1)st generation. generation Probability theory 2011

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

  20. Branching processes- mean and variance of generation size • Consider a branching process for which X0 = 1, and  and  respectively depict the expectation and standard deviation of the offspring distribution. • Then . Probability theory 2011

  21. Branching processes- extinction probability • Let 0 =P(population dies out) and assume thatX0 = 1 • Then where g is the probability generating function of the offspring distribution Probability theory 2011

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

More Related