BCOR 1020 Business Statistics Lecture 9 – February 14, 2008

Overview Chapter 6 – Discrete Distributions –Probability Models –Random Variables –Discrete Distributions –Uniform Distribution*

Chapter 6 – Probability Models Probability Models: A random (or stochastic) process is a repeatable random experiment. Probability can be used to analyze random (or stochastic) processes and to understand business processes. Probability models depict the essential characteristics of a stochastic process to guide decisions or make predictions. –Probability models with well-known properties can be used to describe many stochastic processes.

Chapter 6 – Random Variables Random Variables: A random variable is a function or rule that assigns a numerical value to each outcome in the sample space of a random experiment. –This allows us to perform mathematical operations on all of our experimental variables! Nomenclature: –Capital letters are used to represent random variables (e.g., X, Y). –Lower case letters are used to represent values of the random variable (e.g., x, y).

Chapter 6 – Random Variables Random Variables: A discrete random variable has a countable number of distinct values. –For a discrete categorical variable, the random variable assigns a numerical value to each category in the sample space. –For a discrete numerical variable, the random variable usually assigns each number in the sample space to itself. A continuous random variable has an uncountable (infinite) number of distinct values. –For a continuous variable, the random variable usually assigns each number in the sample space to itself.

Chapter 6 – Random Variables Examples: Suppose we are conducting a survey for customer satisfaction and the response choices are satisfied, uncertain, unsatisfied. If customer response is our variable, we have a discrete categorical variable with sample space S = {satisfied, uncertain, unsatisfied} and we might define the following random variable: –X(satisfied) = 1 –X(uncertain) = 0 –X(unsatisfied) = -1

Chapter 6 – Random Variables Examples: Suppose we are conducting a survey for customer satisfaction and the response choices are satisfied, uncertain, unsatisfied. If we survey 10 customers and the number who say they are satisfied is our variable, we have a discrete numerical variable with sample space S = {0, 1, 2, 3, …, 10} and we might define the following random variable: –X(0) = 0, X(1) = 1, X(2) = 2, X(3) = 3, …,X(10) = 10

Chapter 6 – Discrete Distributions Probability Distributions: A discrete probability distribution is a rule (function) that assigns a probability to each value of a discrete random variable X. To be a valid probability, each probability must be between 0  P( x i )  1 and the sum of all the probabilities for the values of X must be equal to unity.

Chapter 6 – Discrete Distributions Example: Recall from our first lecture on probability, the experiment in which two dice are rolled and we observe their sum. The Venn Diagram of the experiment is pictured… The Sample Space S consists of 36 possible outcomes (6 sides of the 1 st die by 6 sides of the 2 nd ) If our random variable X is the sum, there are eleven possible values of the random variable: x i = 2, 3, 4, …, 12.

Chapter 6 – Discrete Distributions Example (continued): By observing the Venn Diagram we can determine the probability distribution for the sum of the two dice… There is only one way in which the dice will sum to 2 or 12; so We can define the probabilities of all the other possible sums in a similar manner…

Chapter 6 – Discrete Distributions Example (continued): We can come up with a single rule (or function which defines all of our probabilities… Verify that this rule satisfies the conditions of a probability distribution… By Inspection, 0  P(x i )  1 for x = 2, 3, 4, …, 12. Also…

Chapter 6 – Discrete Distributions Expected Value: The expected value, E(X), is a measure of central tendency. For a discrete random variable, the expected value, E(X), is the sum of all X-values weighted by their respective probabilities. If there are n distinct values of X,

Chapter 6 – Discrete Distributions The probability distribution of emergency service calls on Sunday by Ace Appliance Repair is: What is the average or expected number of service calls? x P(x)P(x)P(x)P(x) Total1.00 Example: Service Calls

Chapter 6 – Discrete Distributions The sum of the xP(x) column is the expected value or mean of the discrete distribution. xP(x)P(x)xP(x) Total First calculate x i P(x i ): Example: Service Calls

Chapter 6 – Discrete Distributions This particular probability distribution is not symmetric around the mean  = However, the mean is still the balancing point, or fulcrum.  = 2.75 Because E(X) is an average, it does not have to be an observable point. Example: Service Calls

Clickers On the graveyard shift, the number of patients with head trauma in an emergency room has the following probability distribution: What is the average number of patients with head trauma? A = 0.95B = 1.00 C = 2.25D = 3.25 E = 15.0 x012345Total P(x)

Chapter 6 – Discrete Distributions

If there are n distinct values of X, then the variance of a discrete random variable is: The variance is a weighted average of the dispersion about the mean and is denoted either as  2 or V(X). The standard deviation is the square root of the variance and is denoted . Variance and Standard Deviation:

Clickers On the graveyard shift, the number of patients with head trauma in an emergency room has the following probability distribution: What is the standard deviation of the number of patients with head trauma? A = 1.30B = 1.69 C = 2.25D = 3.25 x012345Total P(x)

Chapter 6 – Discrete Distributions

What is a PDF or CDF? A probability distribution function (PDF) is a mathematical function that shows the probability of each X-value. –Denoted by a lower-case f… A cumulative distribution function (CDF) is a mathematical function that shows the cumulative sum of probabilities, adding from the smallest to the largest X-value, gradually approaching one. –Denoted by an upper-case F…

Chapter 6 – Discrete Distributions Illustrative PDF (Probability Density Function) Cumulative CDF (Cumulative Density Function) Consider the following illustrative histograms: The equations for these functions depend on the parameter(s) of the distribution. What is a PDF or CDF?

Chapter 6 – Discrete Distributions Example: Head Trauma The PDF is given by the table… The CDF is determined by summing values of the p.d.f. up to and including x: x012345Total P(x)

Clickers On the graveyard shift, the number of patients with head trauma in an emergency room has the CDF Find P(1 < X < 4). A = 0.35B = 0.60 C = 0.90D = 0.95

Chapter 6 – Uniform Distribution Characteristics of the Uniform Distribution: The uniform distribution describes a random variable with a finite number of integer values from a to b (the only two parameters). Each value of the random variable is equally likely to occur. Consider the following summary of the uniform distribution:

Chapter 6 – Uniform Distribution Parameters a = lower limit b = upper limit PDF Range a  x  b Mean Std. Dev. Random data generation in Excel =a+INT((b-a+1)*RAND()) Comments Used as a benchmark, to generate random integers, or to create other distributions.

Chapter 6 – Uniform Distribution Example: Rolling a Die The number of dots on the roll of a die form a uniform random variable with six equally likely integer values: 1, 2, 3, 4, 5, 6 PDF for one dieCDF for one die What is the probability of rolling any of these?

Chapter 6 – Uniform Distribution The PDF for all x is: Calculate the standard deviation as: Calculate the mean as: Example: Rolling a Die