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

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  1. Outline • The need for transforms • Probability-generating function • Moment-generating function • Characteristic function • Applications of transforms to branching processes Probability theory 2008

  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 2008

  3. Using transforms to determine the distribution of a sum of random variables Probability theory 2008

  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 2008

  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 2008

  6. The moment generating functionand the Laplace transform Let X be a non-negative random variable. Then Probability theory 2008

  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 2008

  8. The moment generating function- calculation of moments Probability theory 2008

  9. The moment generating function- uniqueness Probability theory 2008

  10. Normal approximation of a binomial distribution Let X1, X2, …. be independent and Be(p) and let Then But Probability theory 2008

  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 2008

  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 2008

  13. The characteristic function- uniqueness For discrete distributions we have For continuous distributions with we have . Probability theory 2008

  14. The characteristic function- calculation of moments If the k:th moment exists we have . Probability theory 2008

  15. Using a normal distribution to approximate a Poisson distribution Let XPo(m) and set Then . Probability theory 2008

  16. Using a Poisson distribution to approximate a Binomial distribution Let XBin(n ; p) Then If p = 1/n we get . Probability theory 2008

  17. Sums of a stochastic number of stochastic variables Probability generating function: Moment generating function: Characteristic function: Probability theory 2008

  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 2008

  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 2008

  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 2008

  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 2008

  22. Exercises: Chapter III 3.1, 3.2, 3.3, 3.7, 3.15, 3.25, 3.26, 3.27, 3.32 Probability theory 2008

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