Probability Theory Modelling random phenomena

Permutations the number of ways that you can order n objects is: n! = n(n-1)(n-2)(n-3)…(3)(2)(1) Definition: 0! = 1 the number of ways that you can choose k objects from n objects in a specific order:

Combinations the number of ways that you can choose k objects from n objects (order irrelevant) is:

are called Binomial Coefficients

Reason: The Binomial Theorem

Binomial Coefficients can also be calculated using Pascal’s triangle

Random Variables Probability distributions

Definition: A random variable X is a number whose value is determined by the outcome of a random experiment (random phenomena)

Examples 1.A die is rolled and X = number of spots showing on the upper face. 2.Two dice are rolled and X = Total number of spots showing on the two upper faces. 3.A coin is tossed n = 100 times and X = number of times the coin toss resulted in a head. 4.A person is selected at random from a population and X = weight of that individual.

5.A sample of n = 100 individuals are selected at random from a population (i.e. all samples of n = 100 have the same probability of being selected). X = the average weight of the 100 individuals.

In all of these examples X fits the definition of a random variable, namely: –a number whose value is determined by the outcome of a random experiment (random phenomena)

Random variables are either Discrete –Integer valued –The set of possible values for X are integers Continuous –The set of possible values for X are all real numbers –Range over a continuum.

Discrete Random Variables Discrete Random Variable: A random variable usually assuming an integer value. a discrete random variable assumes values that are isolated points along the real line. That is neighbouring values are not “possible values” for a discrete random variable Note: Usually associated with counting The number of times a head occurs in 10 tosses of a coin The number of auto accidents occurring on a weekend The size of a family

Examples Discrete –A die is rolled and X = number of spots showing on the upper face. –Two dice are rolled and X = Total number of spots showing on the two upper faces. –A coin is tossed n = 100 times and X = number of times the coin toss resulted in a head.

Continuous Random Variables Continuous Random Variable: A quantitative random variable that can vary over a continuum A continuous random variable can assume any value along a line interval, including every possible value between any two points on the line Note: Usually associated with a measurement Blood Pressure Weight gain Height

Examples Continuous –A person is selected at random from a population and X = weight of that individual. –A sample of n = 100 individuals are selected at random from a population (i.e. all samples of n = 100 have the same probability of being selected). X = the average weight of the 100 individuals.

Probability distribution of a Random Variable

The probability distribution of a discrete random variable is describe by its : probability function p(x). p(x) = the probability that X takes on the value x.

Probability Distribution & Function Probability Distribution: A mathematical description of how probabilities are distributed with each of the possible values of a random variable. Notes: The probability distribution allows one to determine probabilities of events related to the values of a random variable. The probability distribution may be presented in the form of a table, chart, formula. Probability Function: A rule that assigns probabilities to the values of the random variable

Comments: Every probability function must satisfy: 1.The probability assigned to each value of the random variable must be between 0 and 1, inclusive: 2.The sum of the probabilities assigned to all the values of the random variable must equal 1: 3.

Examples Discrete –A die is rolled and X = number of spots showing on the upper face. –Two dice are rolled and X = Total number of spots showing on the two upper faces. x p(x)1/6 x p(x) 1/362/363/364/365/366/365/364/363/362/361/36

Graphs To plot a graph of p(x), draw bars of height p(x) above each value of x. Rolling a die

Rolling two dice

Example In baseball the number of individuals, X, on base when a home run is hit ranges in value from 0 to 3. The probability distribution is known and is given below: PX()the random variable equals 2 p()  Note: This chart implies the only values x takes on are 0, 1, 2, and 3. If the random variable X is observed repeatedly the probabilities, p(x), represents the proportion times the value x appears in that sequence.

A Bar Graph

Note: In all of the examples 1.0  p(x) 

An Important discrete distribution The Binomial distribution Suppose we have an experiment with two outcomes – Success(S) and Failure(F). Let p denote the probability of S (Success). In this case q=1-p denotes the probability of Failure(F). Now suppose this experiment is repeated n times independently.

Let X denote the number of successes occuring in the n repititions. Then X is a random variable. It’s possible values are 0, 1, 2, 3, 4, …, (n – 2), (n – 1), n and p(x) for any of the above values of x is given by:

X is said to have the Binomial distribution with parameters n and p.

Summary: X is said to have the Binomial distribution with parameters n and p. 1. X is the number of successes occurring in the n repetitions of a Success-Failure Experiment. 2.The probability of success is p. 3.

Examples: 1.A coin is tossed n = 5 times. X is the number of heads occuring in the 5 tosses of the coin. In this case p = ½ and x p(x)

Another Example 2.An eye surgeon performs corrective eye surgery n = 10 times. The probability of success is p = 0.92 (92%). Let X denote the number of successful operations in the 10 cases.

Continuous Random Variables

The probability distribution of a continuous random variable is described by its : probability density curve f(x).

i.e. a curve which has the following properties : 1. f(x) is always positive. 2. The total are under the curve f(x) is one. 3. The area under the curve f(x) between a and b is the probability that X lies between the two values.

An Important continuous distribution The Normal distribution The normal distribution (with mean  and standard deviation  ) is a continuous distribution with a probability density curve that is: 1.Bell shaped 2.Centered at the value of the mean  3.The spread is determined by the value of the standard deviation . The points of inflection on the bell curve occur at  -  and  + .

The graph of the Normal distribution    Points of Inflection

Mean and Variance of a Discrete Probability Distribution Describe the center and spread of a probability distribution The mean (denoted by greek letter  (mu)), measures the centre of the distribution. The variance (  2 ) and the standard deviation (  ) measure the spread of the distribution.  is the greek letter for s.

Mean of a Discrete Random Variable The mean, , of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together: Notes: The mean is a weighted average of the values of X. The mean is the long-run average value of the random variable. The mean is centre of gravity of the probability distribution of the random variable

 2 Variance and Standard Deviation Variance of a Discrete Random Variable: Variance,  2, of a discrete random variable x is found by multiplying each possible value of the squared deviation from the mean, (x   ) 2, by its own probability and then adding all the products together: Standard Deviation of a Discrete Random Variable: The positive square root of the variance:

Example The number of individuals, X, on base when a home run is hit ranges in value from 0 to 3.

Computing the mean: Note: is the long-run average value of the random variable is the centre of gravity value of the probability distribution of the random variable

Computing the variance: Computing the standard deviation:

The Binomial distribution 1.We have an experiment with two outcomes – Success(S) and Failure(F). 2.Let p denote the probability of S (Success). 3.In this case q=1-p denotes the probability of Failure(F). 4.This experiment is repeated n times independently. 5.X denote the number of successes occuring in the n repititions.

The possible values of X are 0, 1, 2, 3, 4, …, (n – 2), (n – 1), n and p(x) for any of the above values of x is given by: X is said to have the Binomial distribution with parameters n and p.

Summary: X is said to have the Binomial distribution with parameters n and p. 1.X is the number of successes occurring in the n repetitions of a Success-Failure Experiment. 2.The probability of success is p. 3. The probability function

Example: 1.A coin is tossed n = 5 times. X is the number of heads occurring in the 5 tosses of the coin. In this case p = ½ and x p(x)

Computing the summary parameters for the distribution – ,  2, 

Computing the mean: Computing the variance: Computing the standard deviation:

Example: A surgeon performs a difficult operation n = 10 times. X is the number of times that the operation is a success. The success rate for the operation is 80%. In this case p = 0.80 and X has a Binomial distribution with n = 10 and p = 0.80.

Computing p(x) for x = 1, 2, 3, …, 10

The Graph

Computing the summary parameters for the distribution – ,  2, 

Computing the mean: Computing the variance: Computing the standard deviation: