
Contents
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What is a function?
What is a function?
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What is a random variable?
What is a random variable?
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Can you show an example?
Can you show an example?
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Is a random variable a function?
Is a random variable a function?
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Where do we go from here?
Where do we go from here?
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What is the probability of observing y = 3 heads?
What is the probability of observing y = 3 heads?
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How do we move from the probability of a given value of Y to the probability distribution for all possible values of Y?
How do we move from the probability of a given value of Y to the probability distribution for all possible values of Y?
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Is this an example of a probability distribution?
Is this an example of a probability distribution?
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Is this also a probability mass function?
Is this also a probability mass function?
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What if we had flipped the coin 10 times?
What if we had flipped the coin 10 times?
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Really?
Really?
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OK, what does “binomial” mean?
OK, what does “binomial” mean?
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When do we use binomial probability?
When do we use binomial probability?
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What does the binomial probability mass function look like?
What does the binomial probability mass function look like?
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What notation should I use to describe a binomial process like coin flipping?
What notation should I use to describe a binomial process like coin flipping?
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What is a binomial distribution?
What is a binomial distribution?
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How about the probability of observing 2.5 heads out of 3 coin flips, given that the coin is fair?
How about the probability of observing 2.5 heads out of 3 coin flips, given that the coin is fair?
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What is a parameter?
What is a parameter?
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What are the assumptions of the binomial probability mass function?
What are the assumptions of the binomial probability mass function?
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Are there other probability mass functions besides the binomial?
Are there other probability mass functions besides the binomial?
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What do all of these functions have in common?
What do all of these functions have in common?
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All right then…what is the Bernoulli distribution?
All right then…what is the Bernoulli distribution?
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Likelihood Likelihood
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OK, what exactly is likelihood?
OK, what exactly is likelihood?
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Why is this important?
Why is this important?
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Are there any other key points to bear in mind regarding likelihood?
Are there any other key points to bear in mind regarding likelihood?
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Can we quickly confirm that the likelihoods do not need to sum to 1.0 here?
Can we quickly confirm that the likelihoods do not need to sum to 1.0 here?
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How would this be used in a Bayesian inference problem?
How would this be used in a Bayesian inference problem?
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Can we depict this problem graphically?
Can we depict this problem graphically?
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Can we compare this problem with the authorship problem?
Can we compare this problem with the authorship problem?
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What if we considered all possible hypotheses for p between 0 and 1 instead of just two specific hypotheses?
What if we considered all possible hypotheses for p between 0 and 1 instead of just two specific hypotheses?
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Can we summarize the main points of this chapter?
Can we summarize the main points of this chapter?
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OK, what’s next?
OK, what’s next?
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Cite
Abstract
This chapter focuses on probability mass functions. One of the primary uses of Bayesian inference is to estimate parameters. To do so, it is necessary to first build a good understanding of probability distributions. This chapter introduces the idea of a random variable and presents general concepts associated with probability distributions for discrete random variables. It starts off by discussing the concept of a function and goes on to describe how a random variable is a type of function. The binomial distribution and the Bernoulli distribution are then used as examples of the probability mass functions (pmf’s). The pmfs can be used to specify prior distributions, likelihoods, likelihood profiles and/or posterior distributions in Bayesian inference.
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