Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance and Standard Deviation for the Binomial Distribution 5-5 The Poisson Distribution

Section 5-1 Overview

Overview discrete probability distributions This chapter will deal with the construction of discrete probability distributions by combining the methods of descriptive statistics presented in Chapter 2 and 3 and those of probability presented in Chapter 4. Probability Distributions will describe what will probably happen instead of what actually did happen. Emphasize the combination of the methods in Chapter 2 (descriptive statistics) with the methods in Chapter 3 (probability). Chapter 2 one would conduct an actual experiment and find and observe the mean, standard deviation, variance, etc. and construct a frequency table or histogram Chapter 3 finds the probability of each possible outcome Chapter 4 presents the possible outcomes along with the relative frequencies we expect Page 182 of text

Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect. page 182 of text

Section 5-2 Random Variables

Key Concept This section introduces the important concept of a probability distribution, which gives the probability for each value of a variable that is determined by chance. Give consideration to distinguishing between outcomes that are likely to occur by chance and outcomes that are “unusual” in the sense they are not likely to occur by chance.

Definitions Random variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure Probability distribution a description that gives the probability for each value of the random variable; often expressed in the format of a graph, table, or formula page 183 of text

Example 12 Jurors are to be randomly selected from a population in which 80% of the jurors are Mexican-American. If we assume that jurors are randomly selected without bias, here is an example of a probability distribution depicting these probabilities (let x = number of Mexican-American jurors among 12 jurors). Probabilities that are very small (such as 0.000000123) are represented by 0+. x 1 2 3 4 5 6 7 8 9 10 11 12 P(x) 0+ 0.001 0.003 0.016 0.053 0.133 0.236 0.283 0.206 0.069

Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities. page 184 of text Probability Histograms relate nicely to Relative Frequency Histograms of Chapter 2, but the vertical scale shows probabilities instead of relative frequencies based on actual sample results Observe that the probabilities of each random variable is also the same as the AREA of the rectangle representing the random variable. This fact will be important when we need to find probabilities of continuous random variables - Chapter 5.

Definitions Discrete random variable either a finite number of values or countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they result from a counting process. Example: Count of number of movie patrons Continuous random variable infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions. Example: The measured voltage of a smoke detector battery. This chapter deals exclusively with discrete random variables - experiments where the data observed is a ‘countable’ value. Give examples. Following chapters will deal with continuous random variables. Page 183 of text

Continuous or discrete? The number of eggs a hen lays The count of the number of stats students present in class today The amount of milk a cow produces in one day The measure of voltage for a particular smoke detector battery

Requirements for Probability Distribution P(x) = 1 where x assumes all possible values.  0  P(x)  1 for every individual value of x. Page 185 of text.

Example Does this table describe a probability distribution? Does P(x) = x/3 (where x can be 0, 1, or 2) determine a probability distribution? X 1 2 3 P(x) 0.2 0.5 0.4 0.3

Standard Deviation of a Probability Distribution Mean, Variance and Standard Deviation of a Probability Distribution µ =  [x • P(x)] Mean 2 =  [(x – µ)2 • P(x)] Variance 2 = [ x2 • P(x)] – µ 2 Variance (shortcut)  =  [x 2 • P(x)] – µ 2 Standard Deviation In Chapter 2, we found the mean, standard deviation,variance, and shape of the distribution for actual observed experiments. The probability distribution and histogram can provide the same type information. These formulas will apply to ANY type of probability distribution as long as you have have all the P(x) values for the random variables in the distribution. In section 4 of this chapter, there will be special EASIER formulas for the special binomial distribution. The TI-83 and TI-83 Plus calculators can find the mean, standard deviation, and variance in the same way that one finds those values for a frequency table. With the TI-82, TI-81, and TI-85 calculators, one would have to multiply all decimal values in the P(x) column by the same factor so that there were no decimals and proceed as usual. Page 186 of text.

Roundoff Rule for µ, , and 2 Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ, ,and 2 to one decimal place. If the rounding results in a value that looks as if the mean is ‘all’ or none’(when, in fact, this is not true), then leave as many decimal places as necessary to correctly reflect the true mean. Page 187 of text.

Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = μ + 2σ Minimum usual value = μ – 2σ Page 187-188 of text.

The table below describes the prob. dist The table below describes the prob. dist. for the number of Mexican-Americans among 12 randomly selected jurors in Hidalgo County, Texas. Assuming that we repeat the process of randomly selecting 12 jurors and counting the number of Mexican-Americans each time, find the mean number of Mexican-Americans (among 12), the variance, and the std. dev. X P(x) x*P(x) X-sq X-sq* P(x) 0+ 0.000 1 2 4 3 9 0.001 0.004 16 5 0.003 0.015 25 0.075 6 0.016 0.096 36 0.576 7 0.053 0.371 49 2.597 8 0.133 1.064 64 8.512 0.236 2.124 81 19.116 0. 0.283 2.830 100 28.300 11 0.206 2.266 121 24.926 12 0.069 0.828 144 9.936 Total 9.598 94.054

Identifying Unusual Results Probabilities Rare Event Rule If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct. Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) ≤ 0.05. The discussion of this topic in the text includes some difficult concepts, but it also includes an extremely important approach used often in statistics. Page 189-190 of text. Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer) ≤ 0.05.

Example If 80% of those eligible for jury duty in Hidalgo County are Mexican-American, then a jury of 12 randomly selected people should have around 9 or 10 who are Mexican-American. Is 7 Mexican-American jurors among 12 an unusually low number? Does the selection of only 7 Mexican-Americans among 12 jurors suggest that there is discrimination in the selection process?

E =  [x • P(x)] Definition The expected value of a discrete random variable is denoted by E, and it represents the average value of the outcomes. It is obtained by finding the value of  [x • P(x)]. E =  [x • P(x)] The mean of a discrete random variable is The same as its expected value! Also called expectation or mathematical expectation Plays a very important role in decision theory page 190 of text

Example If you bet $1 in Kentucky’s Pick 4 Lottery game, you either lose $1 or gain $4999 (the winning prize is $5000, but your $1 bet is not returned, so the net gain is $4999). The game is played by selected a four-digit number between 0000 and 9999. If you bet $1 on 1234, what is your expected value of gain or loss?

You do! The random variable x is a count of the number of girls that occur when two babies are born. Construct a table representing the probability distribution, then find its mean and standard deviation.

Recap In this section we have discussed: Combining methods of descriptive statistics with probability. Random variables and probability distributions. Probability histograms. Requirements for a probability distribution. Mean, variance and standard deviation of a probability distribution. Identifying unusual results. Expected value.