Math 362 - Lab #2 - Fall 2002
Introduction
to Probability Theory
Lab 2: Discrete Random Variables
Equipment: a bag containing 12 colored objects of similar shape. The objects
in the bag are only of two possible colors, which we will call Color A and
Color B. There are 8 objects of Color A and 4 objects of Color B. Look at
the contents of your bag and write down which colors are Color A and Color B.
Definitions:
- A random variable is a real-valued function defined on the
sample space S of some experiment.
- A random variable which can take at most a countable number of possible values
is said to be discrete.
In this lab, we will get familiar with some common discrete random variables.
I. The Bernoulli random variable
Consider the experiment of drawing
one object from the bag. If the object is of Color A, we will consider the outcome
to be a success (S). If not, the outcome is a failure (F).
- Assign a probability to each outcome in the sample space of this experiment.
We will denote by p the probability of a success.
- We define the Bernoulli random variable X according to
the following rule:
- X = 1 if the outcome of the experiment is S.
- X = 0 if F occurs.
Explain why X is a discrete random variable.
- Perform the experiment 20 times and record the values of X in the table
below. Each of these experiments is called a Bernoulli trial.
| Experiment # | X (Bernoulli) |
Experiment # | X (Bernoulli) |
| 1 | ... | 11 | ... |
| 2 | ... | 12 | ... |
| 3 | ... | 13 | ... |
| 4 | ... | 14 | ... |
| 5 | ... | 15 | ... |
| 6 | ... | 16 | ... |
| 7 | ... | 17 | ... |
| 8 | ... | 18 | ... |
| 9 | ... | 19 | ... |
| 10 | ... | 20 | ... |
II. The binomial random variable
Consider an experiment which consists
of n independent Bernoulli trials, each of which has the probability p of
being successful. Let the random variable X represent the number of successes
in one experiment. Then X is said to be binomial with parameters
(n, p).
- What are the values that X can possibly take? Is this a finite or infinite
number of values?
- We can consider that 5 consecutive Bernoulli trials in the above table
correspond to one experiment. Record the value of the binomial variable X for
each of the 4 experiments described in the above table. Then perform 2 more experiments
and record the corresponding values of X.
| Experiment # | X (Binomial) |
Experiment # | X (Binomial) |
| 1 | ... | 4 | ... |
| 2 | ... | 5 | ... |
| 3 | ... | 6 | ... |
III. The geometric random variable
Let X be the number of
Bernoulli trials necessary for exactly one success to occur. If p is the
probability of a success, we say that X is a geometric random
variable with parameter p.
- What are the values that X can possibly take? Is this a finite or infinite
number of values?
- Fill out the following table, in which you record the value of X for
10 different experiments. You should again use the information recorded in the table of
Section I above and perform some new experiments if necessary.
| Experiment # | X (Geometric) |
Experiment # | X (Geometric) |
| 1 | ... | 6 | ... |
| 2 | ... | 7 | ... |
| 3 | ... | 8 | ... |
| 4 | ... | 9 | ... |
| 5 | ... | 10 | ... |
IV. The negative binomial random variable
We now consider the experiment
in which we repeat Bernoulli trials until r successes occur. The random variable
X which corresponds to the number of trials performed until the experiment ends
is called a negative binomial random variable, with parameters r
and p.
- What are the values that X can possibly take? Is this a finite or infinite
number of values?
- Use the information recorded in the table of Section I above to fill out the
following table, where X is negative binomial with r = 3.
Perform a few new experiments if necessary.
| Experiment # | X
(Negative binomial) |
Experiment # | X
(Negative binomial) |
| 1 | ... | 4 | ... |
| 2 | ... | 5 | ... |
| 3 | ... | 6 | ... |
V. The hypergeometric random variable
Consider the experiment of drawing
n objects from the bag, without replacement. Let X be the random
variable giving the number of objects of Color A which have been selected. This random
variable is said to be hypergeometric with parameters 8 (the number of
objects of Color A in the bag), 4 (the number of objects of Color B in the bag) and
n (the number of object selected from the bag).
- What are the values that X can possibly take? Is this a finite or infinite
number of values?
- Perform 6 experiments with n = 3 and record the values of X in
the table below.
| Experiment # | X (Hypergeometric) |
Experiment # | X (Hypergeometric) |
| 1 | ... | 4 | ... |
| 2 | ... | 5 | ... |
| 3 | ... | 6 | ... |
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