# 2/6/2023

## What's the difference between events that are independent and disjoint (mutually exclusive)?

• Independent events can both happen, both not happen, or happen separately.
• Disjoint events can only happen separately.

## Random Variables

A discrete random variable X is a random variable which takes on a finite number of values, say x1, x2, …, xk.  P(X=xi) = pi, for i = 1, 2, …, k and the pi must satisfy:

1. pi is in the interval [0, 1] for each i and
2. p1 + p2 + … + pk = 1.

A continuous random variable X can take on all values in an interval of real numbers.  The probability of an event A, P(A), is equal to the area above A and under a probability density curve.  A probability density curve is a function f(x) that satisfies:

1. f(x) is nonnegative for all x and
2. The total area under the curve f(x) is one

## Dealing with a particular value of the random variable:

• For discrete random variables, the equality sign is very important.  In general, the probability that X is at most c is not equal to the probability that X is less than c.
• The probability model for continuous random variables assigns probabilities to intervals of outcomes, not to individual outcomes.  Thus, the probability that X is at most c is equal to the probability that X is less than c.

## Examples of Discrete and Continuous Probability Distributions

• Class Exercises - see handout