# Bayes' Theorem and Probability Distributions

02/10/2021

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## Outline

• The following topics will be covered in this lecture:

• Bayes' theorem
• Random variables
• Probability distributions
• Probability Mass Functions

## Bayes’ Theorem

• Let us suppose that $$A$$ and $$B$$ are events for which $$P(A)\neq 0$$ and $$P(B)\neq 0$$.
• Consider the statement of the multiplication rule, $P(A \cap B) = P(A\vert B) P(B);$
• yet it is also true that, $P(B \cap A) = P(B \vert A) P(A);$
• and $$P( A \cap B) = P(B \cap A)$$ by definition.
• Putting these statements together, we obtain, \begin{align} &P(A\vert B) P(B) = P(B \vert A ) P(A)\\ \Leftrightarrow & P(A \vert B) = \frac{P(B\vert A) P(A)}{ P(B)} \end{align}
• The statement that $P(A \vert B) = \frac{P(B\vert A) P(A)}{ P(B)}$ is known as Bayes' theorem for $$P(B)>0$$.
• This is nothing more than re-writing the multiplication rule as discussed above, but the result is extremely powerful.
• Bayes' theorem wasn’t widely used in statistics for hundreds of years, until advances in digital computers.
• When digital computers became available, many tools became available using Bayes' theorem as the basis.

## Bayes' theorem continued

• Often, Bayes $P(A \vert B) = \frac{P(B\vert A) P(A)}{ P(B)}$ is used as a way to update the probability of $$A$$ when you have new information $$B$$.
• For example, let the events $$A=$$"it snows in the Sierra" and $$B=$$"it rains in my garden".
• I might think there is a $$P(A)$$ prior probability for snow, without knowing any other information.
• $$P(A\vert B)$$ is the posterior probability of snow in the Sierra given rain in my garden.
• If I found out later in the day that there was rain in my garden, I could update $$P(A)$$ to $$P(A\vert B)$$ by multiplying $P(A\vert B) = P(A) \times \left(\frac{P(B\vert A)}{P(B)}\right)$ directly.
• Although this is a simplistic example, this logic is the basis of many weather prediction techniques.

### Bayes' theorem example 1

• EXAMPLE: suppose that 20% of email messages are spam. The word free occurs in 60% of the spam messages. 13% of the overall messages contain the word free.

• Question: How can we use Bayes' theorem,

$P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}$ to compute the probability of a message being spam, given that it includes the word “free”?

• Let the events be
• $$S=$$ “message is spam” $P(S)=0.2$
• $$F=$$ “message contains the word free” $P(F)=0.13$
• We are looking for $$P(S|F)$$
• The probability of a message that has free in it given that is spam is $P(F|S)=0.6$
• From Bayes' theorem $P(S|F)=\frac{P(F|S)P(S)}{P(F)}$
• $P(S|F)=\frac{0.6(0.2)}{0.13}=0.923$

### Bayes' theorem example 2 Courtesy of Montgomery & Runger, Applied Statistics and Probability for Engineers, 7th edition

• EXAMPLE: recall the chips subject to high levels of contamination. The information is summarized in the table on the left.
• Question: How can we use Bayes' theorem, $P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}$ to find the conditional probability of a high level of contamination present, given that a failure occurred?
• Let the events be
• $$H=$$"chip is exposed to high levels of contamination" $P(H)=0.20$
• $$F=$$"product fails"
• Earlier we computed $$P(F)$$ using the total probability rule as $P(F)=P(F|H)P(H)+P(F|H')P(H')=0.024$ with $P(F|H)=0.10 \text{ and } P(F\vert H') = 0.005$
• The probability of $$P(H | F)$$ is determined from Bayes' theorem \begin{align} P(H|F)&=\frac{P(F|H)P(H)}{P(F)} =\frac{0.10(0.20)}{0.024}=0.83\end{align}

## Random Variables Courtesy of Mario Triola, Essentials of Statistics, 6th edition

• The first concept that we will need to develop is the random variable.
• Prototypically, we can consider the coin flipping example from the motivation:
• $$x$$ is the number heads in two coin flips.
• Every time we repeat two coin flips $$x$$ can take a different value due to many possible factors:
• how much force we apply in the flip;
• air pressure;
• wind speed;
• etc…
• The result is so sensitive to these factors that are beyond our ability to control, we consider the result to be by chance.
• Before we flip the coin twice, the value of $$x$$ has yet-to-be determined.
• After we flip the coin twice, the value of $$x$$ is fixed and possibly known.
• Formally we will define:
• Random Variable
A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment.
• Notation
A random variable is denoted by an uppercase letter such as $$X$$. After an experiment is conducted, the measured value of the random variable is denoted by a lowercase letter such as $$x$$