CS433Modeling and Simulation

Lecture 03 – Part 01

Probability Review

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Dr. Anis Koubâa

Al-Imam Mohammad Ibn Saud UniversityAl-Imam Mohammad Ibn Saud University

http://10.2.230.10:4040/akoubaa/cs433/

27 Oct 2008

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Goals for Today

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Review the fundamental concepts of probability

Understand the difference between discrete and continuous random variable

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Topics

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Sample Space Probability Measure Joint Probability Conditional Probability Bayes’ Rule Law of Total Probability Random Variable Probability Distribution Probability Mass Function (PMF) Continuous Random Variable Probability Density Function (PDF) Cumulative Distribution Function (CDF) Mean and Variance

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Probability Review

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Sample space

In probability theory, the sample space or universal sample space

or event space, often denoted S, Ω or U (for "universe"), of an

experiment or random trial is the set of all possible outcomesset of all possible outcomes.

For example, if the experiment is tossing a coin, the sample space is the

set {head, tail}. For tossing a single six-sided die, the sample space is {1,

2, 3, 4, 5, 6}.

In probability theory, an event is a set of outcomes (a subset of the

sample space) to which a probability is assigned. Typically, when the

sample space is finite, any subset of the sample space is an event

(i.e. all elements of the power set of the sample space are defined as

events).

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Probability measure

Every event (= set of outcomes) is assigned a probability, by a

function that we call a probability measure.

The probability of every set is between 0 and 1, inclusive.

The probability of the whole set of outcomes is 1.

If A and B are two events with no common outcomes, then the

probability of their union is the sum of their probabilities.

A single card is pulled (out of 52 cards).

Possible Events:

having a red card (P=1/2);

Having a Jack (P= 1/13);

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Joint Probability

The probability that event P will not happen (=event ~P will happen)

is 1-prob(P).

Event Union (U = OR) Event Intersection (∩= AND)

Joint Probability (A ∩ B): the probability of two events in

conjunction. That is, it is the probability of both events together.

Independent Events: Two events A and B are independent if

Example

The probability to have a red card and which is a Jack card

p=p(red) * p(jack) = 1/2 * 1/13

p A B p A p B p A B

p A B p A p B

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Conditional Probability

Conditional Probability p(A|B) is the probability of some event A, given the occurrence of some other event B.

If A and B are independent,

then and If A and B are independent, the conditional probability of A, given B

is simply the individual probability of A alone; same for B given A. p(A) is the prior probability; p(A|B) is called a posterior probability. Once you know B is true, the universe you care about shrinks to B.

|p A B

p A Bp B

|

p B Ap B A

P A

|p A B p A |p B A p B

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Bayes' rule

We know that

Using Conditional Probability definition, we have

The Bayes rule is:

||

p B A p Ap A B

p B

p A B p B A

| |p A B P B p B A P A

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Law of Total Probability

In probability theory, the law of total probability

means

the prior probability of A, P(A), is equal to

the expected value of the posterior probability

of A.

That is, for any random variable N,

where p(A|N) is the conditional probability of A given N.

|p A E p A N

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Law of Total Probability

Law of alternatives: The term law of total

probability is sometimes taken to mean the law of

alternatives, which is a special case of the law of

total probability applying to discrete random variables.

if { Bn : n = 1, 2, 3, ... } is a finite partition of a probability

space and each set Bn is measurable, then for any

event A we have

or, alternatively,

nn

p A p A B

| nn

p A p A B p B

Five Minutes Break

You are free to discuss with your classmates about the previous slides, or to refresh a bit, or to ask questions.

Administrative issues• Groups Formation• Choose a “class coordinator”

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A random variable

Random Variable is also know as stochastic variable. A random variable is not a variable. It is a function. It maps

from the sample space to the real numbers. random variable is defined as a quantity whose values are

random and to which a probability distribution is assigned. More formally: a random variable is a measurable

function from a sample space to the measurable space of possible values of the variable

Example A coin is tossed ten times. The random variable X is

the number of tails that are noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable.

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Probability Distribution

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The probability distribution of a discrete

random variable is a list of probabilities

associated with each of its possible values.

It is also sometimes called the probability

function or the probability mass function

(PMF) for discrete random variable.

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Probability Mass Function (PMF)

Formally the probability distribution or probability

mass function (PMF) of a discrete random variable X is a function that gives the probability p(xi) that the random variable equals xi, for each value xi:

It satisfies the following conditions: 0 1 ip x

1 ii

p x

i ip x P X x

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Continuous Random Variable

A continuous random variable is one which takes an infinite number of possible values.

Continuous random variables are usually measurements.

Examples include height, weight, the amount of sugar in an orange, the time required to run a mile.

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Distribution function aggregates

For the case of continuous variables, we do not want to ask what the probability of "1/6" is, because the answer is always 0...

Rather, we ask what is the probability that the value is in the interval (a,b).

So for continuous variables, we care about the derivative of the distribution function at a point (that's the derivative of an integral). This is called a probability density function (PDF).

The probability that a random variable has a value in a set A is the integral of the p.d.f. over that set A.

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Probability Density Function (PDF) The Probability Density Function (PDF) of a

continuous random variable is a function that can be integrated to obtain the probability that the random variable takes a value in a given interval.

More formally, the probability density function, f(x), of a continuous random variable X is the derivative of the cumulative distribution function F(x):

Since F(x)=P(X≤x) it follows that

d

f x F xdx

b

a

F b F a P a X b f x dx

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Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is a function giving the probability that the random variable X is less than or equal to x, for every value x.

Formally the cumulative distribution function F(x) is

defined to be:

,

x

F x P X x

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Cumulative Distribution Function (CDF)

For a discrete random variable, the cumulative distribution function is found by summing up the probabilities as in the example below.

For a continuous random variable, the cumulative distribution function is the integral of its probability density function f(x).

,

( ) ( )

i i

i ix x x x

x

F x P X x P X x p x

b

a

F a F b P a X b f x dx

Two Minutes Break

You are free to discuss with your classmates about the previous slides, or to refresh a bit, or to ask questions.

Administrative issues• Groups Formation• Choose a “class coordinator”

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Cumulative Distribution Function (CDF) Example

Discrete case Discrete case : Suppose a random variable X has the following probability mass function p(xi):

The cumulative distribution function F(x) is then:

xi 0 1 2 3 4 5

p(xi) 1/32 5/32 10/32 10/32 5/32 1/32

xi 0 1 2 3 4 5

F(xi) 1/32 6/32 16/32 26/32 31/32 32/32

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Discrete Distribution Function

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Mean and variance

The expected value (or population mean) of a random variable indicates its average or central value.

The expected value gives a general impression of the behavior of some random variable without giving full details of its probability distribution (if it is discrete) or its probability density function (if it is continuous).

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n

X i ii

E X x p x

X E X x f x dx

Expectation of discrete random variable X

Expectation of continuous random variable X

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Mean and variance

Example When a die is thrown, each of the possible faces

1, 2, 3, 4, 5, 6 (the xi's) has a probability of 1/6 (the p(xi)'s) of showing. The expected value of the face showing is therefore:

µ = E(X) = (1 x 1/6) + (2 x 1/6) + (3 x 1/6) + (4 x 1/6) + (5 x 1/6) + (6 x 1/6) = 3.5

Notice that, in this case, E(X) is 3.5, which is not a possible value of X.

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Mean and variance The variance is a measure of the 'spread' of a distribution

about its average value. Variance is symbolized by V(X) or Var(X) or σ2.

Whereas the mean is a way to describe the location of a distribution, the variance is a way to capture its scale or degree of being spread out. The unit of variance is the square of the unit of the original variable

The Variance of the random variable X is defined as:

where E(X) is the expected value of the random variable X. The standard deviation is defined as the square root of

the variance, i.e.:

2 22 2XV X E X E X E X E X

2X X V X s