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)!]*pk*(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 x1, x2, …, xk with probabilities p1, p2, …, pk the mean (or expected value) of X is given by µ=E[X] = x1*p1 + x2*p2 + ... + xk*pk.
- Variance of X
- If X is a discrete random variable taking values x1, x2, …, xk with probabilities p1, p2, …, pk the variance of X is given by Var[X] = (x1-µ)2*p1 + (x2-µ)2*p2 + (xk-µ)2*pk.
- 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.