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Conditional probability mass function

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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- 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 - rules

Probability theory 2008

Calculation of expected valuesthrough 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 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

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

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

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 errorof the best linear predictor

Theorem:

Proof: …….

Ordinary linear regression

Probability theory 2008

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

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 | < h, where h > 0

- Determines the probability function of X uniquely
- Adding independent variables corresponds to multiplying their moment generating functions

Probability theory 2008

The moment generating functionand 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 X1, X2, …. be independent and Be(p) and let

Then

.

Probability theory 2008

Distributions for which the moment generating function does not exist

Let X = eY, where YN( ;)

Then

and

.

Probability theory 2008

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

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 generating function- uniqueness

For discrete distributions we have

For continuous distributions with

we have

.

Probability theory 2008

The characteristic function generating 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 XPo(m) and set

Then

.

Probability theory 2008

Using a Poisson distribution to approximate a Binomial distribution

Let XBin(n ; p)

Then

If p = 1/n we get

.

Probability theory 2008

Sums of a stochastic number of stochastic variables distribution

Probability generating function:

Moment generating function:

Characteristic function:

Probability theory 2008

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 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 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 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

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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