# 2/8/2023

## Problem Session - Discussion of Discrete and Continuous Probability Distributions Lab

## Binomial Experiments (A generalization of the discrete RV lab problem on the number of patients who experience side effects)

- Each random trial can result in one of only two possible outcomes. This is called a Bernoulli trial.
- We collect data from Bernoulli trials satisfying the following:
- there are n trials;
- the n trials are independent; and
- the probability of “success” remains constant from trial to trial.

- Binomial Distribution - If X is the number of successes in n independent Bernoulli trials, then P(X=k)=n!/[k!*(n-k)!]*p
^{k}*(1-p)^{n-k}, for k=0, 1, ..., n.
- We will learn how to use RStudio to make these calculations easier in Chapter 4.

## The mean (expected value) and standard deviation of
a random variable

- Mean (or Expected Value) of X
- If X is a discrete random variable taking values x
_{1}, x_{2}, …, x_{k} with probabilities p_{1}, p_{2}, …, p_{k} the mean (or expected value) of X is given by µ=E[X] = x_{1}*p_{1} + x_{2}*p_{2} + ... + x_{k}*p_{k}.

- Variance of X
- If X is a discrete random variable taking values x
_{1}, x_{2}, …, x_{k} with probabilities p_{1}, p_{2}, …, p_{k} the variance of X is given by Var[X] = (x_{1}-µ)^{2}*p_{1} + (x_{2}-µ)^{2}*p_{2} + (x_{k}-µ)^{2}*p_{k}.
- The standard deviation of X is the square root of the variance.

## Properties of means and variances

## Conditional Probability and Bayes Theorem - (see Section3.2_Conditional_Probability_Bayes_Rule.pdf in our !Class-Notes Google Drive folder.

## Class Activity - Applications of Bayes' Theorem

## We will begin Chapter 4 on Monday so please read Section 4.1 for class on Monday.

## We will have a quiz on Sections 3.1, 3.3, 3.4, and 3.5 on Friday. Once again, the quiz will be open book, open notes.

##