Random Variables , Distributions and Expectation 1 Random Variables

We’ve used probablity to model a variety of experiments, games, and tests. Throughout, we have tried to compute probabilities of events. We asked, for example, what is the probability of the event that you win the Monty Hall game? What is the probability of the event that it rains, given that the weatherman carried his umbrella today? What is the probability of the event that you have a rare disease, given that you tested positive? But one can ask more general questions about an experiment. How hard will it rain? How long will this illness last? How much will I lose playing 6.042 games all day? These ques­ tions are fundamentally different and not easily phrased in terms of events. The problem is that an event either does or does not happen: you win or lose, it rains or doesn’t, you’re sick or not. But these questions are about matters of degree: how much, how hard, how long? To approach these questions, we need a new mathematical tool.