Business and Finance College Principles of Statistics Eng. Heba Hamad 2008

Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University

Chapter 5 Introduction Discrete probability distributions – Chapter 5 –Variables have specific values (e.g. 0, 1, 2.5) Continuous probability distributions – Chapter 6 –Variables have range of values (e.g. 0-1, ) Will use tables of probabilities in order to minimize amount of computation required

Chapter 5 Discrete Probability Distributions Random Variables Discrete Probability Distributions Expected Value and Variance Binomial Distribution

A random variable is a numerical description of the outcome of an experiment. Random Variables A discrete random variable may assume either a finite number of values or an infinite sequence of values. A continuous random variable may assume any numerical value in an interval or collection of intervals.

Random Variables Question Random Variable x Type Family size x = Number of dependents reported on tax return Discrete Distance from home to store x = Distance in miles from home to the store site Continuous Own dog or cat x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) Discrete

Discrete Random Variable Example Example: experiment of tossing a coin –x is a discrete random variable (1 for heads and 2 for tails) –Outcome probability: f (x) = ½

Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) Example: JSL Appliances Discrete random variable with a finite number of values

Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... Example: JSL Appliances n Discrete random variable with an infinite sequence of values We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

Examples of Discrete Random Variables

Continuous Random Variable Examples ExperimentRandom Variable (x) Possible Values for x Bank tellerTime between customer arrivals x >= 0 Fill a drink container Number of millimeters 0 <= x <= 200 Construct a new building Percentage of project complete as of a date 0 <= x <= 100 Test a new chemical process Temperature when the desired reaction take place 150 <= x <= 212

The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. We can describe a discrete probability distribution with a table, graph, or equation. Discrete Probability Distributions

The probability distribution is defined by a probability function, denoted by f ( x ), which provides the probability for each value of the random variable. The required conditions for a discrete probability function are: Discrete Probability Distributions f ( x ) > 0  f ( x ) = 1

Table of a Discrete Probability Distribution for a Roll of a Die xf(x) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 f(x) = 1/6, for x=1, 2, 3, 4, 5, 6

Graph of a Discrete Probability Distribution for a Roll of a Die

Toss of a Coin F(x) = ½, for x=1, 2

n a tabular representation of the probability distribution for TV sales was developed. n Using past data on TV sales, … Number Units Sold of Days x f ( x ) /200 Discrete Probability Distributions

Values of Random Variable x (TV sales) Probability Discrete Probability Distributions Graphical Representation of Probability Distribution

Discrete Uniform Probability Distribution The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula. The discrete uniform probability function is f ( x ) = 1/ n where: n = the number of values the random variable may assume the values of the random variable are equally likely

Toss of a Coin f(x) = ½, for x =1, 2 f(1) = ½ f(2) = ½ sum of f(x) = 1.0

Roll of a Die F (x) = 1/6, for x = 1, 2, 3, 4, 5, 6 f(1) = 1/6 f(2) = 1/6 f(3) = 1/6 f(4) = 1/6 f(5) = 1/6 f(6) = 1/6 sum of f(x) = 1.0

Example: Dicarlo Motors Consider the sales of automobiles at Dicarlo Motors we define x = no of automobiles sold during a day Over 300 days of operation, sales data shows the following:

Example: Dicarlo Motors No. of automobiles sold No. of days Total300

Example: Dicarlo Motors xf(x) Total1.00

Example