MATH 214 Lab 6: Random Variable Applications
As usual, open Word and set up your lab report.
Before beginning, please read the information below to “review” some various random variables.
Characteristics of the Random Variables in this lab:
Binomial
1. An experiment is repeated a fixed number of times, n.
2. The variable is the number of times one possible outcome, called Success, occurs in these n trials.
3. The probability of Success, p, remains constant from one trial to another.
4. Each trial is independent of the others.
5. Parameters: n, p,  = np, and 2 = np(1 – p).
Poisson
1. The experiment consists of counting the number of times a certain event, Success, occurs during a given
unit of time or in a given area or volume (or weight, distance, or any other continuous unit of
measurement). This count is the value of the random variable.
2. The probability that Success occurs in a given unit of time, area, volume, etc., is the same for all units.
3. The number of Successes that occur in one unit of time, area, volume, etc., is independent of the number
of Successes that occur in other units.
4. The mean (or expected) number of events in each unit is denoted by the Greek letter lambda, .
5. Parameters:  =  = 2.
Hypergeometric
1. The experiment consists of randomly drawing a pre-specified number, n, of elements without
replacement from a set of N elements, r of which are Successes, and N – r of which are Failures.
2. The value of the random variable is the number of Successes drawn.
3. Parameters: N, r, n,  = nr/N, and 2 =(r(N – r)n(N – n))/(N2(N – 1)).
Geometric
1. An experiment with two possible outcomes, Success or Failure, is repeated until Success occurs.
2. The variable is the number of trials required until the first Success.
3. The probability of Success, p, remains constant from one trial to another.
4. Each trial is independent of the others.
5. Parameters: p,  = 1/p, and 2 = (1 – p)/p2.
Exponential
1. The experiment consists of measuring the time (or distance, area, volume, weight, or any other
continuous unit of measurement) until a certain event, Success, occurs.
2. The probability that Success occurs in a given unit of time, area, volume, etc., is the same for all units.
3. The number of Successes that occur in one unit of time, area, volume, etc., is independent of the number
of Successes that occur in other units.
4. The mean (or expected) waiting time is denoted by the Greek letter theta,, which is the reciprocal of the
mean, , of the corresponding Poisson random variable. Theta is also known as the scale.
5. Parameters:  =  = 1/λ = 
For each of the random variables described in A-F below, determine whether it is a binomial, Poisson,
hypergeometric, geometric, or exponential random variable. Also identify or estimate as many parameters
(mean, standard deviation etc.) of its probability distribution as you can.
Experiments:
A. Roll a six-sided die 5 times. The variable X is the number of fours observed in those five rolls.
B. Take a bowl of beads. Choose one color. Find and record the probability of randomly selecting a bead of
the color you picked by counting the total number of beads in the bowl and the number of beads of the
color you chose. Randomly pick one bead at a time (with replacement) until the selected color appears.
The variable X is the number of drawings required.
C. Select 3 cards (without replacement) at random from a set of 13. The 13 cards consist of A, 2, 3, …, 10,
Jack, Queen, and King, all of the same suit (Hearts, Clubs, Diamonds, or Spades) from a standard 52card deck of playing cards. The variable X is the total number of 10’s, Jacks, Queens and Kings in those
3 cards.
D. Select 3 cards (with replacement) at random from a set of 13. The 13 cards consist of A, 2, 3, …, 10,
Jack, Queen, and King, all of the same suit (Hearts, Clubs, Diamonds, or Spades) from a standard 52card deck of playing cards. The variable X is the total number of 10’s, Jacks, Queens and Kings in those
3 cards.
E. Observe a particular entrance to the Henson (or any other classroom) building. The variable N is the
number of people passing through that entrance in a one minute period.
F. Observe a particular entrance to the Henson (or any other classroom) building. The variable T is the time
between arrivals of a group of people at that entrance.
Hints on data collection for E. and F.:
 As each individual reaches the appointed spot, record the time, in minutes and seconds, since you began
your observations. Don’t try to keep track of the variables in E and F while you are collecting data.
Rather, process the data as you record it back in the lab after it is all collected.
 Try to avoid collecting data near the change of classes.
Collect a sample of at least 20 observations, from the population described in each experiment.
To save time, work in pairs and share samples with the other person. Enter the data into MINITAB.
For each experiment:
(a) Identify the theoretical distribution that models the experiment, its parameters as listed in the
description of each distribution above, and graph the distribution. Note, if parameters are unknown,
use the estimates from your sample in constructing the theoretical graph.
In MINITAB:
Click Graph, Select Probability distribution plot, select view single
Specify the distribution and its parameter(s).
(b) Calculate the descriptive statistics of your sample—in particular, the mean, variance, and range.
(c) Display your sample data in a histogram.
(d) Compare the distribution of your sample data to the theoretical distribution. Be sure to compare the
sample statistics (mean, variance, etc.) to the population parameters (parameters of the theoretical
distribution).