Probability theory 2008 Conditional probability mass function  Discrete case  Continuous case

Probability theory 2008 Conditional probability mass function - examples  Throwing two dice  Let Z 1 = the number on the first die  Let Z 2 = the number on the second die  Set Y = Z 1 and X = Z 1 +Z 2  Radioactive decay  Let X = the number of atoms decaying within 1 unit of time  Let Y = the time of the first decay

Probability theory 2008 Conditional expectation  Discrete case  Continuous case  Notation

Probability theory 2008 Conditional expectation - rules

Probability theory 2008 Calculation of expected values through conditioning  Discrete case  Continuous case  General formula

Probability theory 2008 Calculation of expected values through conditioning - example  Primary and secondary events  Let N denote the number of primary events  Let X 1, X 2, … denote the number of secondary events for each primary event  Set Y = X 1 + X 2 + … + X N  Assume that X 1, X 2, … are i.i.d. and independent of N

Probability theory 2008 Calculation of variances through conditioning Variation in the expected value of Y induced by variation in X Average remaining variation in Y after X has been fixed

Probability theory 2008 Variance decomposition in linear regression

Probability theory 2008 Proof of the variance decomposition We shall prove that It can easily be seen that

Probability theory 2008 Regression and prediction Regression function: Theorem: The regression function is the best predictor of Y based on X Proof: Function of X

Probability theory 2008 Best linear predictor Theorem: The best linear predictor of Y based on X is Proof: ……. Ordinary linear regression

Probability theory 2008 Expected quadratic prediction error of the best linear predictor Theorem: Proof: ……. Ordinary linear regression

Probability theory 2008 Martingales The sequence X 1, X 2,… is called a martingale if Example 1: Partial sums of independent variables with mean zero Example 2: Gambler’s fortune if he doubles the stake as long as he loses and leaves as soon as he wins

Probability theory 2008 Exercises: Chapter II 2.6, 2.9, 2.12, 2.16, 2.22, 2.26, 2.28 Use conditional distributions/probabilities to explain why the envelop-rejection method works

Probability theory 2008 Transforms

Probability theory 2008 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 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

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

Probability theory 2008 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 The moment generating function - calculation of moments

Probability theory 2008 The moment generating function - uniqueness

Probability theory 2008 Normal approximation of a binomial distribution Let X 1, X 2, …. be independent and Be(p) and let Then.

Probability theory 2008 Distributions for which the moment generating function does not exist Let X = e Y, where Y  N(  ;  ) Then and.

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

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

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

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

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

Probability theory 2008 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

Probability theory 2008 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:

Probability theory 2008 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.

Probability theory 2008 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

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