345 ACTIVITY 7: Using Probability Mass and Cumulative Distribution Functions to Compute the Probability of Events

WHY:

This activity is intended to clarify the relationship between discrete sample spaces, discrete random variables, probability mass functions, and cumulative distribution functions, with a focus on computing probabilities using these concepts. We will use cumulative distribution functions often in this course to organize different approaches to computing probability. The real importance of the cumulative distribution and probability mass function is their ability to describe the behavior of a population from a plot of sample data.

LEARNING OBJECTIVES:

1. Discover the relationship between discrete sample space, random variable, probability mass function, and cumulative distribution function.
2. Review the computation of probability of events.
3. Discover how other groups interact during discovery learning.

CRITERIA:

1. Quality of the answers to the Critical Thinking Questions.
   • Completeness
   • Clarity
   • Depth
2. Quality of the report on the other groups' process.

INFORMATION:

Sample Space

The set of possible outcomes of a statistical experiment. It is discrete if it contains finitely or countably many elements. It is continuous if it contains an interval of real numbers.

Random Variable

A function whose domain is a sample space and whose range is a subset of the real numbers. It is discrete if its domain is a discrete sample space, and continuous if its domain is a continuous sample space. Random Variables can be 1-1 (map a single sample space element to a unique real number) or many-one (map a whole event to a real number).

Probability Mass Function

For each value x_i in the range of a discrete random variable X we can compute the probability f(x_i) of the sample space event that maps into x_i. The pairs (x_i, f(x_i)) define a probability mass function which follows the rules

• 0 <=f(x_i) <= 1
• f(x₁) + f(x₂ + ... +f(x_n = 1
• f(x_i) = P(X = x_i).

The cumulative distribution function F(x) associated with f(x) is defined for all real numbers as F(x) = f(x₁) + f(x₂ + ... + f(x_i) where x_i <= x < x_i+1.

Probability Histogram

The graph of a step function formed from the probability mass function f(x) by using (x_i+1 + x_i)/2 as the step points and f(x_i+1) as the height of the step. Choose the first step point to be x₁ - (x₁ + x₂)/2 and the last to be x_n + (x_n-1 + x_n)/2.

Computing Discrete Probability

In order to more easily compute probabilities and graph results of a statistical experiment, we map the sample space into the real numbers {x₁, x₂, ...} using a random variable X with probability mass function f(x) and cumulative distribution function F(x). To compute P(a <= X <= b), find the sum of f(x_i) from x_i = a to x_i = b. To compute P(a < X <= b), find F(b) - F(a). Note that P(a <= X <= b) may be different from P(a < X <= b) if a is one of the x_i.

RESOURCES:

1. Sections 3.1, 3.2: Probability and Statistics for E & S
2. Counting Methodology
3. 30 minutes

PLAN:

1. Choose roles if you have not already done so.
2. Work through the model, verifying each step.
3. Answer the critical thinking questions.

MODEL:

A disk jockey selects 4 records at random from a collection consisting of 4 jazz records, 2 classical records, and 3 polka records. The sample space can be represented as {R1R2R3R4} where Ri is either a jazz, classical or polka record. Let X be the number of classical records chosen. The following table applies:

Possible Values for X:       0                1             2

 # ways to choose x     7  6  5  4       2  7  6  5     2  1  7  6
                        ----------	-----------	----------
 classical records          4!               3!            2! 2!   

# ways to choose any    9  8  7  6       9  8  7  6     9  8  7  6
		        ----------	 ----------	----------
 4 classical records        4!              4!		    4!

P(X = x)  (I.e. f(x))      5/18            5/9             1/6

     The cumulative distribution function associated with  f(x)  is:
                    {   0         x < 0
                    |   5/18      0 <= x < 1
          F(x) =    |  15/18      1 ó x < 2
                    {    1        x >= 2
 

CRITICAL THINKING QUESTIONS:

These will be provided in class.

Return to the List of Activities