SESSION 31 & 32

Last Update14th April 2011

Discrete Probability Distributions

Lecturer: Florian BoehlandtUniversity: University of Stellenbosch Business SchoolDomain: http://www.hedge-fund-analysis.net/pages/v

ega.php

Learning Objectives

All measures for grouped data: 1. Data types revisited2. Probability trees – with and without

replacement3. Binomial distribution

Random Variable

A random variable is a function or rule that assigns a number to each outcome of an experiment. We can distinguish between continuous and discrete random variables.A probability distribution is a table, formula, or graph that describes the values of a random variable. Generally, an uppercase letter describes the name and the lowercase counterpart the value of the random variable:

Quantitative Data

Data

ContinuousDiscrete

Continuous probability distribution

functions (most notably normal,

student-t, F, Chi-Squared)

Binomial and Poisson

distribution

Data variables can only assume certain values and are collected typically by counting observations

Data variables can assume any value within a range (real numbers) and are collected by measurement.

Binomial distribution

1. The binomial experiment consists of a fixed number of trials n.

2. Each trial has two possible outcomes. One is labeled success, the other failure.

3. The probability of success is p. The probability of failure is 1 – p.

4. The trials are independent (i.e. the outcome of one trial does not effect the outcome of any other)

Example: Tossing a coin n times and record the results

Bernoulli Process

If the properties 2, 3 and 4 from the previous slide are satisfied, we denote each trial as a Bernoulli process. The random variable of a binomial variable is defined as the number of successes in the n trials (binomial random variable).The binomial random variable can take on the values 0, 1, 2, …, n. The random variable is discrete. A probability tree represents the events in an experiment by lines. It can be used to calculate joint probabilities.

Probability Tree

S

S

SS

S

S

S

F

F

F

F

F

F

F

Trial 1 Trial 3Trial 2 Trial nSFSFSF

SF

.

.

.

. . .

Probability Tree – Without Replacement

First Choice Second Choice

A statistics course has 7 male and 3 female students. The lecturer wants to select two students at random for a research project. What is the probability that the two students are female?

M = 7 /10

F = 3 /10

M = 6 / 9

M = 7 / 9

F = 3 / 9

F = 2 / 9

Joint Probabilities

Probability Tree – With Replacement

First Choice Second Choice

The professor’s replacement is teaching the next two classes while the professor is sick. The lecturer selects one student at random and picks on him or her to answer questions. What is the probability that the two students chosen are female?

M = 7 /10

F = 3 /10

M = 7 / 10

M = 7 / 10

F = 3 / 10

F = 3 / 10

Joint Probabilities

Probability Tree – cont.

Note that one differentiates between experiments with and without replacement. In this case, once a student is drawn from the sample, he/she is not replaced in the next draw. Texas Hold’em Poker is another typical example of experiments without replacement. This implies that the trials are not independent (i.e. violates property 4. of binomial experiments).

In experiments with replacement, the associated probabilities do not change from trial to trial. The trials are independent.

Probability of Sequence

Assuming the trial are independent, we can use the probability of sequence formula to calculate the probability of x successes in n trials:

Where p denotes the probability of success.

Combinatorial Formula

Using a probability tree, there may be a number of branches (combinations) that yield x successes and n-x failures. For example, there are two ways to produce a one success and one failure in two trials: SF and FS. To count the number of branches, the combinatorial formula is employed (with C as the number of combinations for a specific n and x):

Binomial Probability Distribution

The combination of the probability of sequence and the combinatorial formula yields the binomial probability distribution (i.e. the probability of x successes in a binomial experiment with n trials and probability of success = p):

Example: Stats Class – With Replacement

What is the probability for the combination Male / Female to be picked in two consecutive classes by the substitute professor?

Solution

1. Define one event as success (e.g. x: female)2. Determine the probability for success:

F / (F + M) = 3/ 10 (conversely, M / (F + M) = 7 / 10 is the probability for failure, i.e. male)

3. The other elements are:n = 2 (for two trials)x = 1 (for the number of required successes)

Solution cont.

4. Enter all elements into the formula:

You can use your calculator to calculate the first part of the formula (combinations) using the nCr key.

Exercise: Statistics Exam

Consider a student taking a statistics course. The student has not prepared at all for an upcoming statistics quiz. The quiz consists of ten multiple choice questions. Each question has five possible answers of which only one is correct. Assume that the student is willing to guess randomly to answer each question.a)What is the probability that the student gets no answers correct?b)Two answers correct?c)Ten answers correct? d)What is the chance that the student will fail the quiz assuming 50% correct answers are required to pass?