9/24/2021

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 x₁, x₂, …, x_k with probabilities p₁, p₂, …, p_k the mean (or expected value) of X is given by µ=E[X] = x₁*p₁ + x₂*p₂ + ... + x_k*p_k.
• Variance of X
• If X is a discrete random variable taking values x₁, x₂, …, x_k with probabilities p₁, p₂, …, p_k the variance of X is given by Var[X] = (x₁-µ)²*p₁ + (x₂-µ)²*p₂ + (x_k-µ)²*p_k.
• The standard deviation of X is the square root of the variance.

Properties of means and variances

Class Activity - Means and Variances of Discrete Random Variables

Bayes' Theorem

Class Activity - Applications of Bayes' Theorem

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