Download Handout 9 Supplements material in Section 3.4 of L-G

Handout 9
Supplements material in Section 3.4 of L-G
Equality of RV’s
Consider and as two random variables. There are lots of ways one might think of these
as the same random variable. One way might be if both and are defined on the same sample
space , and they are exactly the same function. They are then called (identically) equal random
variables. A second (and more common) type of equality is if the distribution functions are the
same. This is best done in an example. Let be a random variable which records the number of
heads in 2 tosses of a coin. can take on the values
with probabilities
. Now
consider another random variable, , which is determined by the throw of a sided dice.
is
defined as
if the dice shows 1
if the dice shows 2 or 3
if the dice shows 3
Now, can take on the values
with probabilities
. Clearly, and are not
the same random variable. They are not even defined on the same sample space. Equally clearly,
once we abstract from the sample space to the random variable, both
and
have the same
behavior. and are considered to be identically distributed random variables and we write this
as
Discrete Random Variables
We have already seen a couple of discrete random variables in Chapter 2 without using the
words “Random Variable”. The Binomial Random Variable corresponds to the number of successes out of independent repetitions of an experiment, where the probability of success in a
single experiment is . The probability function of the Binomial random variable is
for
. The cdf of the Binomial rv cannot be written in a closed-form expression.
The Geometric Random Variable corresponds to the number of repetitions of the experiment it
takes to achieve the first success. The probability function of the Geometric rv is
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36-217: Probability Theory and Random Processes
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. The cdf of the Geometric rv can be written in a closed form expression and it is
A very important property of the Geometric random variable is the memoryless property:
for all
. This states that if a success has not occurred in the first trials, then the probability of having to repeat the experiment more times in order to get a success is the same as the
probability of initially performing the experiment times in order to get a success.
The geometric random variable arises in applications where one is interested in time until the
occurrence of an event or between two events.
Examples on the Geometric Distribution
Example 9.1 An oil prospector will drill a succession of holes in a given area to find a productive
well. The probability that he is successful on a given trial is 0.2.
(a) What is the probability that the third hole drilled is the first that yields a productive well?
(b) If the prospector can only afford o drill at most ten wells, what is the probability that he fails
to find a productive well?
Solution. Let’s define the random variable which denotes the number of holes needed to be
drilled in order to get the first productive well. Then will follow a Geometric Distribution with
parameter
.
(a)
the third hole is the first success
(b)
first 10 holes are nonproductive
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36-217: Probability Theory and Random Processes
Fall 1997
The Negative Binomial Random Variable
The Negative Binomial Distribution
A random variable with a negative binomial distribution originates from a context that is very
similar to the one that leads to the geometric distribution. Again we focus on independent and
identical trials each of which results on one of two outcomes, “success” or “failure”. The probability of success is and stays the same from trial to trial. The Geometric distribution handles
the case where we are interested in the number of trials until the first success occurs. What if we
are interested in the number of the trial on which the second, third of fourth success occurs? The
distribution that applies to the random variable , equal to the number of the trial on which the th
success occurs (
) is the Negative Binomial Distribution. Let’s, now, find the probability
distribution of .
the th trial is the one in which the th success occurs
successes in the first
trials)
and a success on the th trial
where
Example 9.2 A geological study indicates that an exploratory oil well drilled in a particular region
should strike oil with probability 0.2. Find the probability that the third oil strike comes on the fifth
well drilled.
Solution. Let denote the number of the trial on which the third oil strike occurs. Then it is
reasonable to assume that has negative binomial distribution with parameters
and
.
Thus
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36-217: Probability Theory and Random Processes
Fall 1997
The Poisson Distribution
Consider an experiment where you set up a Geiger counter and count the number of clicks (say
particle emissions) in a fixed period of time. There is clearly no limit to the number of clicks
(for practical purposes there is a limit, but let’s ignore that), so the range of the number of clicks
is
. Other examples include the number of packets on an Ethernet in some time period,
the number of cars passing by Forbes and Morewood for a given length of time etc.
The random variable that denotes such counts as above is said to be the Poisson random variable. The probability distribution function of the Poisson random variable is given by the following
expression:
The parameter
will turn out to be important–it is actually the mean of the distribution.
Example 9.3 There are two entrances to a parking lot. Cars arrive at entrance I according to
a Poisson distribution at an average of three an hour, and a entrance II according to a Poisson
distribution at an average of four per hour. What is the probability that three cars arrive at the
parking lot in a given hour? (Assume that the numbers of cars arriving at the two entrances are
independent)
Solution. Let be the number of cars entering the tunnel in a given 2-minute period.
Poisson distribution with
. Then
has a
There are
trials now, and let the number of times (out of 10) that the event
occurs. Then follows a Binomial distribution with parameters
and
. Thus
the probability of interest becomes:
Next time: The Poisson rv (continued) and Continuous rvs
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