Transcript Statistics 400

Statistics 270 - Lecture 10
• Last day/Today: Discrete probability distributions
• Assignment 3: Chapter 2: 44, 50, 60, 68, 74, 86, 110
Common Discrete Probability Distributions
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There are several probability distributions that describe a large
variety of random phenomena
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Will consider 5 of these:
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2.
3.
4.
5.
Discrete Uniform
Bernoulli
Binomial
Hypergeometric
Poisson
Discrete Uniform Distribution
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Have seen this already
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Random variable X has k possible outcomes, each equally likely
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pmf:
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Mean and Variance:
Bernoulli Distribution
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Have seen this already
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Random variable X has 2 possible outcomes
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pmf:
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Mean and Variance:
Binomial Distribution
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Count the number of successes in n independent Bernoulli trials
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Binomial Experiment:
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Have n trials, where n is fixed in advance of the experiment
Each trial results in one of two possible outcomes (success or failure)
The outcomes are independent
The probability of success is constant
Binomial Distribution
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Count the number of successes in n independent Bernoulli trials
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Binomial Experiment:
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Have n trials, where n is fixed in advance of the experiment
Each trial results in one of two possible outcomes (success or failure)
The outcomes are independent
The probability of success is constant
Binomial Distribution
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Let X denote the number of successes in n independent Bernoulli
trials
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Then the rv, X, is said to be a binomial rv
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pmf
Binomial Distribution
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Mean:
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Variance:
Example
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A baseball player has a 300 batting average
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What is the expected number of hits in 25 at bats?
Example
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According to CTV News, the 2006 Federal Election results were:
Party
Conservative
Liberal
NDP
Bloc Quebecois
Independent
Green
Other
% of Popular Vote
36.3
30.2
17.5
10.5
0.6
4.5
0.4
Example
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Ten voters from across the country are randomly selected and the
number of of Conservative voters is counted
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Is this a Binomial experiment?
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What is the probability that 6 of them voted for the Conservatives?
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What is the expected number of Conservative votes in such a
sample?
Hyper-geometric Distribution
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Count the number of successes in n trials from a population with N
individuals and M successes
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Assumptions:
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Have n trials, where n is fixed in advance of the experiment
Population has N individuals (finite population)
There are two possible outcomes (success and failure) and there are
M successes in the population
A sample of n individuals is taken WITHOUT replacement
Hyper-geometric Distribution
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Let X denote the number of successes in a sample of size n,
without replacement
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Then the rv, X, is said to be a hyper-geometric rv
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pmf
Hyper-geometric Distribution
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Mean:
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Variance:
Example (Chapter 3 - 64)
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A digital camera comes in either a 3 or 4 mega-pixel version
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A store receives 15 cameras and 6 are the 3 mega-pixel version
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Suppose 5 of these are randomly selected and stored behind the
counter
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Let X denote the number of 3 mega-pixel cameras stored behind
the counter
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Compute P(X=2) and P(X <=2)