Outline
Binomial experiment
Binomial random variable
Exercises
Chapter 3 - Lecture 5
The Binomial Probability Distribution
Andreas Artemiou
October 12th, 2009
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Binomial experiment
Experiment
Examples
Binomial random variable
Definition
Distribution
Moments and moment generating function of a Binomial
Random Variable
Exercises
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Experiment
Examples
Definition
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A binomial experiment is one that satisfies the four
following requirements
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The experiment consists of a sequence of n smaller experiment
called trials, where n is fixed in advance of the experiment
Each trial can result in one of the same two possible outcomes
success or failure.
The trials are independent, so that the outcome on any
particular trial does not influence the outcome on any other
trial
The probability of success is constant from trial to trial.
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Experiment
Examples
Examples
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Toss of a coin
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Roll a die, with for example S = {1, 2} and F = {3, 4, 5, 6}
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Birth of a child
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Experiment
Examples
Examples - a small issue
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Suppose in Stat 318 there are 50 students, 44 men and 6
women. I want to select a committee of three to
communicate all the requests to the instructor. The first one
is the President, the second one is the vice president and the
third one is just a member.
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Suppose in Penn State there are 50000 students, with only
10000 being women. I want to select a committee of three to
communicate all requests to the President of the University.
The first one is the President, the second one is the vice
president and the third one is just a member.
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Are the above experiments, binomial experiments?
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Are there any issues with them?
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Experiment
Examples
Rule of thumb
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If we sample without replacement from a large sample, we will
consider that it is a binomial experiment if our sample size is
less than 5% of the population of interest.
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Definition
Distribution
Moments and moment generating function of a Binomial Random
Definition
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When we have a binomial experiment consisting of n trials, the
binomial random variable X associated with this experiment
is defined as the number of successes among the n trials
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Definition
Distribution
Moments and moment generating function of a Binomial Random
Binomial Distribution
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Every random variable has a distribution
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A binomial random variable has the Binomial distribution
which is affected by two parameters, the number of trials n
and the probability of success.
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The distribution is denoted as B(n, p)
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Definition
Distribution
Moments and moment generating function of a Binomial Random
Example
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If X ∼ B(n, p) and we want to find the P(X = x) we use the
following formula:
  n p x (1 − p)n−x , x = 0, 1, . . . , n
x
P(X = x) =

0
, otherwise
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Definition
Distribution
Moments and moment generating function of a Binomial Random
Binomial Tables
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Imagine that you have a random variable X ∼ B(25, 0.2).
How would you find P(X < 13)?
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Definition
Distribution
Moments and moment generating function of a Binomial Random
Expected value and Variance
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If X ∼ B(n, p) then:
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E (X ) = np
var(X ) = np(1 − p)
Example: If X ∼ B(7, 0.35) find the expected value the
variance and the standard deviation of X .
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Definition
Distribution
Moments and moment generating function of a Binomial Random
Moment Generating Function
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If X ∼ B(n, p) then MX (t) = (pe t + 1 − p)n
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Proof?
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Find E (X ) and var(X ) of a Binomial random variable using
the moment generating function
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution
Outline
Binomial experiment
Binomial random variable
Exercises
Exercises
I
Section 3.5 page 132
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Exercises 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
72, 73, 75, 76, 77, 78, 79
Andreas Artemiou
Chapter 3 - Lecture 5 The Binomial Probability Distribution