Chapter 16 Random Variables Random Variables l These

Chapter 16 Random Variables

Random Variables l These are variables which are extracted from random phenomena. l Let’s say we wanted to know the number of heads found on 3 flips of a fair coin. l HTH and THH are two different outcomes. l They are, however, the same number of heads.

Random Variables So, to clarify, the outcomes are converted into the number of heads, and the number of heads is what we consider on this random variable. l We get the following chart: l # of Heads Probability 0 . 125 1 . 375 2 . 375 3 . 125

What is the point? l Random variables refer to numbers taken from random events. l These can be the outcomes, or else counts of outcomes. l Now that these are numbers, though, instead of categorical, we can do even more calculations with them.

Mean and Standard Deviation l Woohoo! l Seriously, though, mean and standard deviation of a random variable are easy. l The formulas look intimidating, but it is a smokescreen.

Mean l l l Now we will also called it expected value. The expected value is the mean. The mean is the expected value. The mean may still be written µ, but now it might also be written E(X) or E(Whatever), for the expected value. To find expected value, we multiply each outcome by its probability and add them all up.

Example Time! If we have the following chart: l 0 x. 5 = 0, 1 x. 3 =. 3, 2 x. 15 =. 3, 3 x. 05 =. 15. l 0 +. 3 +. 15 =. 75 l So the mean is. 75 l X 0 1 2 3 P(X). 5. 3. 15. 05

Variance l l l The standard deviation is the square root of the variance. We will find the variance first. Then, we will square root that answer if we want the standard deviation. The formula is almost as easy. We will multiply the square of each outcome with the probability, add them up, and then subtract the mean squared.

Example Time We will use our previous example. l The mean was. 75. l In order to find the variance, we will need to square each outcome, or each X value. l X 0 1 2 3 P(X). 5. 3. 15. 05

Example Time l l l So our new table looks like this: So 0 x. 5 = 0, 1 x. 3 =. 3, 4 x. 15 =. 6, 9 x. 05 =. 45. 0 +. 3 +. 6 +. 45 = 1. 35. 752 = 0. 5625 1. 35 – 0. 5625 = 0. 7875 The variance is 0. 7875 X 0 1 2 3 P(X). 5. 3. 15. 05 X 2 0 1 4 9

Example Time l The standard deviation is the square root of the variance, so:

Mean and Standard Deviation l It is seriously that easy. l Not even kidding. l The book uses a harder formula for variance, so feel free to use that one instead. l Hehehe.

Mean and Variance l If we take a random variable and multiply it by a number, we can find the new mean by multiplying the old mean by that number. l We multiply the standard deviation by that number. l We multiply the variance by the square of that number.

Mean and Variance l When we add two random variables, we add their means and we add their variances. l When we subtract two random variables, we subtract their means and we add their variances. l That’s right! We add variances in both cases.

Mean and Variance If we know the standard deviation instead of the variance, we must square it to get the variance. l Only after we have squared can we add the variances together. l Let’s look at some examples! l For those of you that are not here, you will miss these examples, as they are being done on the board. l

Super Example Time Activate! First, we will use this data to find the mean and standard deviation. l Then we will focus on the variance as well. l X 0 1 2 3 P(X). 5. 3. 15. 05