But suppose we want to find a particular value for y, at which the probability is, say, 0.90? In other words, we want to figure out the following:

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1 More on distributions, and some miscellaneous topics 1. Reverse lookup and the normal distribution. Up until now, we wanted to find probabilities. For example, the probability a Swedish man has a brain weight less than 1,325 gm: Pr{Y < 1,325} But suppose we want to find a particular value for y, at which the probability is, say, 0.90? In other words, we want to figure out the following: Pr{Y < y} = 0.90 We want to find the value of y equivalent to an area of 0.90 (or 90%). In other words, what value of y has 90% of the population below it? We also call these percentiles: The 90 th percentile has 90% of the values below y. The 64 th percentile has 64% of the values below y. and so on. An example from testing: What is the 90th percentile on the GRE (Graduate Record Exam)? (Aside: the GRE is an exam most students need to take if they go into graduate school; it's similar in some ways to things like the SAT, MCAT, etc.). We want to know what score on the GRE corresponds to the 90th percentile. So, for example, if your score is the same as the 90 th percentile, that means you did better than 90% of students taking the GRE - pretty good! Let's try an example using the body mass index (bmi). According to the CDC, the average bmi for men was about 26.6, with a standard deviation of about 4.5 (that last figure is a bit of an estimate). Let's try to find the 80th percentile for men and assume that the figures above represent the population parameters (so μ = 26.6 and σ = 4.5.

2 Let's draw a picture to figure out what we want: So here s how to do it. Remember that our normal distribution table gives us the area (= probability) below a number (z) that we look up. Now we want to do things backwards; we want to find a number that goes with a specific probability: We want to find z in the following: Pr{Z < z} = 0.80 Since we know the probability (area, or percentile) that we want, we look in the table (not on the sides of the table). We look for the number closest number to.80. This turns out to be Now we read the number off the sides and get In other words, we now have: Pr{Z < 0.84} = 0.80 or z = 0.84 This means 80% of the area of our normal curve is below Now the last step is to convert this back to serum cholesterol levels.

3 Remember that z = y We plug in our z, μ and σ and solve for y. y = z Now we can use this to get our y: y = = And we conclude that 80% of men have a bmi less than (Incidentally, what's the bmi cutoff for being overweight? for being obese? These numbers are a little scary). 2. Other distributions (just a few comments): There are many, many other distributions other than just the binomial or the normal. Some, like the binomial, are discrete, others, like the normal, are continuous. Here are just a few examples: Some discrete examples: a) Poisson A typical application might be to model rare events with no upper limit. For example: The number of people in line at a grocery store - usually lines are not too bad, but at peak times (try shopping at a grocery store before a snow storm) it can go WAY up. The number of parasites in a host. Note that there is no upper limit (n is not a parameter here) Here is the formula: f ( y) = e μ μ y y! We can answer questions like what is the probability that a person has three parasites? If we know, for example, that the average number of parasites a person has is 4, then we figure out the probability that someone has 6 parasites as follows: Pr {Y = 6} = e = ! The Poisson distribution is used a lot, and we'll have a few homework problems using it,

4 b) Uniform but for the most part our class won't make much use of it. We've already used this quite a bit when we were calculating the probability of getting a particular number on a single dice. It's simply given by: f y = 1 N Where N is simply the number of outcomes (since all outcomes are equally likely). This is a discrete uniform. There's also a continuous uniform distribution. Some continuous distributions: c) t, χ 2, F 3. A little more on parameters and estimates We won t discuss these right now, but we'll see them again (except the F distribution) later in the semester. The formulas for these are very messy; if you're really interested, check out Wikipedia. Parameters may or may not be directly related to the mean and variance of a distribution. For example: For the normal, the parameters are the mean and standard deviation (μ and σ). Very convenient! For the binomial, the parameters are n and p. But what are the mean and standard deviation for the binomial (they exist!)? If we know n and p (like we do in some of the examples we ve been talking about: coin tosses, beans, etc.): μ = np σ = (np(1 p)) Does this make sense? Well, for the mean: If we toss a coin 10 times, what s the mean for heads? 10 x 0.5 = 5 Yes, that should be pretty obvious. Incidentally, if we didn t know p (we often don t), we would estimate this in the usual

5 way: Sum up the number of heads in each 10 tosses, divide by 10. And our sample mean estimates our actual mean. What about σ? This is more difficult to understand, so let's just take it as a given. But how did we know that μ = np, or σ = (np(1 p))? Or in other words, how is a mean (or standard deviation) defined (mathematically)? Depends: For a discrete distribution it is: n μ = i=0 y i f ( y i ) Here f(y) is our distribution, for example, the binomial. You could do some really messy math and you'd find that: n μ = i=0 y i ( n y i ) py i (1 p) n y i = np (An easier way to see is just to try it out - for example: Plug in.5 and 3 for p and n (that might be tossing a coin three times) Let Y = # of heads (so y would go from 0 to 3) Finally plug everything in to the sum above, and you'll get 1.5). For a continuous distribution it is: = y f y dy Here, again, f(y) is our distribution, fore example the normal. Again, if you do some nasty math you ll find that if you plug the normal distribution in for f(y) you ll get μ. (Yes, this does seem circular, but it's not). Some comments:

6 You might have heard of expected values. The expected value of a random variable is defined this way - simply put, the expected value of a random variable is the mean (μ). For continuous distributions you need calculus, so I don t expect you to know this definition for continuous distributions. The third and fourth editions define expected values just a little differently, but it comes out the same. The above is a little more formal. The third edition and fourth editions also have more information on this (this was mostly missing in the second edition). There are similar equations for the variance (σ 2 ), but they're even messier than the above, so we'll ignore them. Finally, some trivia (you can ignore this if you want). There are probability distributions that do not have a mean (or variance). If you try to plug these distributions into the above equation(s), you'll discover that there is no answer. If you're really interested, look up the Cauchy distribution in Wikipedia (the Cauchy distribution is also programmed into R). Finally: you should remember that the mean (μ) of a population (or distribution) is not the same as a sample mean ( y ). We use the sample mean to estimate the actual mean.