Introduction

What is probability?

Probability is the chance that a given event will occur. Examples

Probability that it will rain tomorrow. Probability that you will win the lottery.

There is often no certain answer so we assign a number between 0 and 1. Where 0 is certain “no” and 1 is certain “yes”.

In all probabilistic problems we have a random experiment (the drawing of lottery ticket), a set of outcomes (number of sequences in lottery), The probabilities assigned (1/N, N = total number of

sequences)

Why studying probability theory?

Probability theory provides us with the ability to predict the behavior of random phenomena in the “long run”. To the extent that this information is useful, probability can serve as a valuable tool for assessment and decision making.

Example: rainfall prediction

We may never be able to predict with certainty the outcome of an experiment, we are nonetheless, able to predict “averages”.

Annual summer rainfall

Rhode Island Arizona

RANDOM PROCESS Random process is a collection of random variables (annual

rainfall); this is often used to represent the evolution of some random value, or system, over time.

Random process can be characterized in some probabilistic manner.

Undesirablerandom waveform (noise)

Desirable random signals

Example 1: Heart Rate VariabilityAccording to the National Heart Lung and Blood Institute, 5 million Americans have heart failure, and 300,000 die from it each year.The costs for these patients are very significant: Total costs for treating heart failure in the US are estimated to be $34.8 billion in 2008.

Hospitalization Rate (per 1000 people) by Age Group for Heart Failure 1979-2004

Applying probability theory and statics we can prevent deaths.

Example 2: FOREX MARKETIn the spot Forex market, almost $2 trillion is traded daily, making it the largest and most liquid market in the world.

How to determine the correct direction of upcoming changes? How to open profitable position for trade? How to close position with maximal profit?

CADJPY One day

Types of probability Discrete (the number of experimental outcomes is finite, of finite

but countable) and continuous. Discrete case: Number of persons at a business location that are

talking on the phone from 9:00 AM and 9:10AM Continuous case: The length of time a particular caller is on the

phone during that time period. Outcomes may be anywhere from 0 to T minutes. Now number of outcomes is infinite.

Probabilistic modeling

The number of telephone callers can be modeled assigning a probability p to each person being on the phone anytime in the given T-minute interval and assuming that whether one or more persons are on the phone does not affect the probability of others being on the phone.

• Coin tossing analogy• Person is on the phone is associated with head. • Person is not on the phone is associated with tails

• Probability models are simplified approximations to reality.

Probabilistic modeling (CONT)

• If there are N = 4 persons in the office, then the experimental outcomes is likened to 4 coin tosses.

• Asking for the probability that 3 persons are on the phone by determining the probability of 3 heads out of 4 coin tosses.

• The solution to this problem is given by

where

Probabilistic modeling (CONT)

if p = 0.75 and N=4 a compilation of the probabilities is shown below.

• The probability that 3 people are on the phone is 0.42.

Probabilistic modeling (CONT) The coin toss model will be poor, if the assumptions are invalid.

Different persons have different probabilities p. The probability of one person being on the phone is affected by

whether his neighbor is on the phone (neighbors tend to talk to each other), i.e., the events are not “independent”.

The probability p changes over time (later in the day there is less phone activity due to fatigue).

Probabilistic modeling (CONT)

• For the modeling of the length of time T a caller is on the phone it is not appropriate to ask for the probability p is exactly T = 5min. It will be zero. Instead we inquire as to the probability that T will be between 5 and 6 minutes.

• For continuous outcomes a frequently used probabilistic model is the Gaussian of “bell” – shaped curve.

Analysis versus computer simulation

Experiment # 1 2 3 … N=1000

# of heads 4 2 1 2

• Then, to determine the probability of 3 heads we use a relative frequency interpretation of probability to yield

• For many practical problems it is not possible to determine pT(t).• However, if we know a model for the random phenomenon, then we

may carry out the experiment a large number of times to obtain an approximate probability.

• To determine the probability of 3 heads in 4 tosses of a coin with probability of heads being p=0.75, we toss the coin four time and count the number of heads.

Analysis versus computer simulation

To implement this approach we need a method to simulate the toss of a coin. In MATLAB a number in the interval (0,1) can be produced with the statement x = rand (1,1).

A computer simulation of the number of persons in the office on the telephone can be implemented with the MATLAB code