1. Types of Random Variables Discrete random variables are ones that have a finite or countable number of possible outcomes (like number of heads when flipping several coins). Continuous random variables are ones that have an infinite or uncountable number of possible outcomes (like your exact speed on the highway, or how far someone jumps)

2. Range Rule of Thumb Yet again, we consider an outcome “unusual” if falls more than two standard deviations from the mean. Maximum usual value: Minimum usual value:

3. Important Consideration Consider guessing the answers on a 100 question true/false test. Is unusual to get 55 questions correct? What are we really asking? P(exactly 55 correct) = 0.0485 P(55 or more correct) = 0.184

4. Important Consideration Although getting exactly 55 correct is unusual, getting 55 answers correct is not considered an unusual event because it is not an unusually high outcome – since the likelihood of getting 55 or more correct is not particularly small. In general, we consider an outcome x: unusually high if P(x or more) is very small unusually low if P(x or less) is very small.

5. Expected Value Remember we said that for discrete random variables, the mean,, is also called the expected value.

6. Example: Basic roulette bet: 38 spaces, bet $1 on one number. If that number comes up, you get $36. Otherwise, you lose. EventxP(x)x. P(x) Win$351/38$35/38 Lose-$137/38-$37/38 (usually we round the mean to one more decimal place than x had, but in this case, we would lose too much meaning if we did)

7. Example: So our expected value of -$0.053 says that on each round, we expect to lose about 5 cents. Of course, on each round we either win or lose, but the expected value tells us what we expect to have happen on average. So if we played 1000 rounds, we would expect to lose about $53.

8. Homework 4.2: 9, 11, 13, 17