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

A

discrete random variableX is a random variable which takes on a finite number of values, say x_{1}, x_{2}, …, x_{k}. P(X=x_{i}) = p_{i}, for i = 1, 2, …, k and the p_{i}must satisfy:

- p
_{i}is in the interval [0, 1] for each i and- p
_{1}+ p_{2}+ … + p_{k}= 1.A

continuous random variableX 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:

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

- 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.

- Class Exercises - see handout