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Conditional probability mass function. Discrete case Continuous case. 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

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Conditional probability mass function l.jpg
Conditional probability mass function

  • Discrete case

  • Continuous case

Probability theory 2008


Conditional probability mass function examples l.jpg
Conditional probability mass function- examples

  • Throwing two dice

    • Let Z1 = the number on the first die

    • Let Z2 = the number on the second die

    • Set Y = Z1 and X = Z1+Z2

  • 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 l.jpg
Conditional expectation

  • Discrete case

  • Continuous case

  • Notation

Probability theory 2008


Conditional expectation rules l.jpg
Conditional expectation - rules

Probability theory 2008


Calculation of expected values through conditioning l.jpg
Calculation of expected valuesthrough conditioning

  • Discrete case

  • Continuous case

  • General formula

Probability theory 2008


Calculation of expected values through conditioning example l.jpg
Calculation of expected values through conditioning- example

  • Primary and secondary events

    • Let N denote the number of primary events

    • Let X1, X2, … denote the number of secondary events for each primary event

    • Set Y = X1 + X2 + … + XN

    • Assume that X1, X2, … are i.i.d. and independent of N

Probability theory 2008


Calculation of variances through conditioning l.jpg
Calculation of variances through conditioning

Average remaining

variation in Y

after X has

been fixed

Variation in the

expected value of Y

induced by

variation in X

Probability theory 2008



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Proof of the variance decomposition

We shall prove that

It can easily be seen that

Probability theory 2008


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Regression and prediction

Regression function:

Theorem:The regression function is the best predictor of Y based on X

Proof:

Function of X

Function of X

Probability theory 2008


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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 l.jpg
Expected quadratic prediction errorof the best linear predictor

Theorem:

Proof: …….

Ordinary linear regression

Probability theory 2008


Martingales l.jpg
Martingales

The sequence X1, X2,… 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


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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


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Transforms

Probability theory 2008


The probability generating function l.jpg
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 l.jpg
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


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The moment generating functionand the Laplace transform

Let X be a non-negative random variable. Then

Probability theory 2008


The moment generating function examples l.jpg
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 l.jpg
The moment generating function- calculation of moments

Probability theory 2008


The moment generating function uniqueness l.jpg
The moment generating function- uniqueness

Probability theory 2008


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Normal approximation of a binomial distribution

Let X1, X2, …. be independent and Be(p) and let

Then

.

Probability theory 2008


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Distributions for which the moment generating function does not exist

Let X = eY, where YN( ;)

Then

and

.

Probability theory 2008


The characteristic function l.jpg
The characteristic function not exist

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


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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


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The characteristic function generating function- uniqueness

For discrete distributions we have

For continuous distributions with

we have

.

Probability theory 2008


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The characteristic function generating function- calculation of moments

If the k:th moment exists we have

.

Probability theory 2008


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Using a normal distribution to approximate a Poisson distribution

Let XPo(m) and set

Then

.

Probability theory 2008


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Using a Poisson distribution to approximate a Binomial distribution

Let XBin(n ; p)

Then

If p = 1/n we get

.

Probability theory 2008


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Sums of a stochastic number of stochastic variables distribution

Probability generating function:

Moment generating function:

Characteristic function:

Probability theory 2008


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Branching processes distribution

  • 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


Generating function of a branching processes l.jpg
Generating function of a branching processes distribution

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


Branching processes mean and variance of generation size l.jpg
Branching processes distribution- 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


Branching processes extinction probability l.jpg
Branching processes distribution- 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


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Exercises: Chapter III distribution

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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