Download Chapter 8 The Binomial and Geometric Distributions

Chapter 8
The Binomial and Geometric Distributions
8.1 The Binomial Distribution
8.2 The Geometric Distribution
In practice, we frequently encounter experimental
situations where there are two outcomes of interest. For
example, tossing a coin or shooting a free throw.
In this chapter, we will explore two important classes of
distributions (binomial and geometric) and learn some of
their properties. Everything we’ve learned so far about
probability and random variables will be used.
When presented with an experimental setting, it is
important to recognize it as a binomial setting or a
geometric setting or neither.
So, how can you tell?
Binomial Setting
• In a binomial setting, X= number of
successes is called a binomial random
variable, and the probability distribution
of X is called a binomial distribution.
Binomial distributions are an important class of discrete
probability distributions, but pay attention – not all counts
have binomial distributions. Let’s look at Exercise 8.1 on
page 441.
Calculating Binomial Probabilities
Let’s look at exercises 8.9 and 8.11 on page 449.
Example 8.15 – Geometric Variable
An experiment consists of rolling a single die. The event of interest is rolling a
3; this event is called a success. The random variable is defined as X = the
number of trials until a 3 occurs. To verify this is a geometric setting, note that
rolling a 3 will represent a success, and rolling any other number will represent
a failure. The probability of rolling a 3 on each roll is the same: 1/6. The
observations are independent. A trial consists of rolling the die once. We roll
the die until a 3 appears. Since all of the requirements are satisfied, this
experiment describes a geometric setting.
Using this setting, let’s calculate some probabilities.
X = 1:
X = 2:
X = 3:
P(X = 1) = P(success on first roll) = 1/6
P(X = 2) = P(success on second roll)
= P(failure on first roll and success on second roll)
= P(failure on first roll) Χ P(success on second roll)
= 5/6 Χ 1/6
P(X = 3) = P(failure on first roll) Χ P(failure on second roll) Χ P(success
on third roll)
= 5/6 Χ 5/6 Χ 1/6
Continue the process, and the pattern suggests the following principle:
A probability distribution table for the geometric random variable is strange
because it never ends; the number of table entries is infinite.
You can use the rule above to construct the table.
You’ll notice that the probabilities are the terms of a geometric sequence.
Also, please note that the sum of the probabilities must still add to 1.
Here is a graph of the distribution of X from example 8.15. Notice that the
probability distribution histogram is strongly skewed to the right with a peak
at the leftmost value. This will always be the case for a geometric
distribution.
Exercise 8.39 – page 468
Suppose we have data that suggest that 3% of a company’s hard disk
drives are defective. You have been asked to determine the probability
that the first defective hard drive is the fifth unit tested.
(a) Verify that this is a geometric setting. Identify the random variable;
that is, write X = number of
and fill in the blank.
What constitutes a success in this situation?
(b) Answer the original question: What is the probability that the first
defective hard drive is the fifth unit tested?
(c) Find the first four entries in the table of the pdf for the random
variable X.
What about the probability that it takes more than a certain number of trials to
achieve success? This can be found as follows:
Calculating Mean and Standard
Deviation of a Binomial Random
Variable
Note: These short formulas are good ONLY
for binomial distributions!!!!
Calculating Mean and Standard
Deviation of a Geometric Random
Variable
•If you’re flipping a fair coin, how many times would you expect to have to flip the
coin in order to observe the first head?
•If you’re rolling a die, how many times would you expect to have to roll the die in
order to observe the first 3?
Normal Approximation of
Binomials
• A useful fact– as the number of trials n gets larger,
the binomial distribution gets close to a Normal
distribution. When n is large, we can use Normal
probability calculations to approximate hard-tocalculate binomial probabilties. The Normal
approximation is most accurate for p values close to
0.5.
Calculating Binomial
Probabilities
(using a graphing calculator)
Additional Examples
Exercise 8.19 on page 455
Exercise 8.33a on page 462
Exercise 8.47ab on page 475
Exercise 8.49 on page 475