Chapter 6 Discrete Probability Distributions

Definitions Random variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure 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 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 page 201 of Elementary Statistics, 10th Edition

EXAMPLE. Distinguishing Between Discrete and EXAMPLE Distinguishing Between Discrete and Continuous Random Variables Determine whether the following random variables are discrete or continuous. State possible values for the random variable. The number of light bulbs that burn out in a room of 10 light bulbs in the next year. (b) The number of leaves on a randomly selected Oak tree. (c) The length of time between calls to 911. Discrete; x = 0, 1, 2, …, 10 Discrete; x = 0, 1, 2, … Continuous; t > 0

Definitions 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 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 201 of Elementary Statistics, 10th Edition

EXAMPLE A Discrete Probability Distribution The table to the right shows the probability distribution for the random variable X, where X represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01

Rules for Probability Distribution P(x) = 1 where x assumes all possible values.  0  P(x)  1 for every individual value of x. Page 203 of Elementary Statistics, 10th Edition

EXAMPLE Identifying Probability Distributions Is the following a probability distribution? x P(x) 0.16 1 0.18 2 0.22 3 0.10 4 0.30 5 0.04

A probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable.

Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities. page 202 of Elementary Statistics, 10th Edition 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 6.

EXAMPLE Drawing a Probability Histogram P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 Draw a probability histogram of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit.

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  =  [(x – µ)2 • P(x)] Standard Deviation In Chapter 3, 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 204 of Elementary Statistics, 10th Edition

EXAMPLE Computing the Mean of a Discrete Random Variable Compute the mean of the probability distribution to the right, which represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01

E =  [x • P(x)] Definition Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X), of the random variable. 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)]. Also called expectation or mathematical expectation Plays a very important role in decision theory page 208 of Elementary Statistics, 10th Edition E =  [x • P(x)]

EXAMPLE Computing the Expected Value of a Discrete Random Variable A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one year term life insurance policy to a 49-year-old female for $530. According to the National Vital Statistics Report, Vol. 47, No. 28, the probability the female will survive the year is 0.99791. Compute the expected value of this policy to the insurance company. x P(x) $530 0.99791 $530 – $250,000 = -$249,470 1- 0.99791 = 0.00209 Survives Does not survive E(X) = 530(0.99791) + (-249,470)(0.00209) = $7.50

EXAMPLE. Computing the Variance and Standard Deviation EXAMPLE Computing the Variance and Standard Deviation of a Discrete Random Variable x P(x) 0.06 1 0.58 2 0.22 3 0.10 4 0.03 5 0.01 Compute the variance and standard deviation of the following probability distribution which represents the number of DVDs a person rents from a video store during a single visit. x P(x) 0.06 -1.43 2.0449 0.122694 1 0.58 -0.91 0.8281 0.480298 2 0.22 -1.27 1.6129 0.354838 3 0.1 -1.39 1.9321 0.19321 4 0.03 -1.46 2.1316 0.063948 5 0.01 -1.48 2.1904 0.021904

Section 6.2 The Binomial Probability Distribution Next: Section 6.2 The Binomial Probability Distribution