Chapter 6 Section 1

Chapter 6Section 1 Discrete Random Variables

6 1 2 3 4 5 Chapter 6 – Section 1 • Learning objectives • Distinguish between discrete and continuous random variables • Identify discrete probability distributions • Construct probability histograms • Compute and interpret the mean of a discrete random variable • Interpret the mean of a discrete random variable as an expected value • Compute the variance and standard deviation of a discrete random variable

Chapter 6 – Section 1 • A randomvariable is a numeric measure of the outcome of a probability experiment • Random variables reflect measurements that can change as the experiment is repeated • Random variables are denoted with capital letters, typically using X (and Y and Z …) • Values are usually written with lower case letters, typically using x (and y and z ...)

Chapter 6 – Section 1 • Examples • Tossing four coins and counting the number of heads • The number could be 0, 1, 2, 3, or 4 • The number could change when we toss another four coins • Examples • Tossing four coins and counting the number of heads • The number could be 0, 1, 2, 3, or 4 • The number could change when we toss another four coins • Measuring the heights of students • The heights could change from student to student

Chapter 6 – Section 1 • A discreterandomvariable is a random variable that has either a finite or a countable number of values • A finite number of values such as {0, 1, 2, 3, and 4} • A countable number of values such as {1, 2, 3, …} • A discreterandomvariable is a random variable that has either a finite or a countable number of values • A finite number of values such as {0, 1, 2, 3, and 4} • A countable number of values such as {1, 2, 3, …} • Discrete random variables are designed to model discrete variables (see section 1.2) • Discrete random variables are often “counts of …”

Chapter 6 – Section 1 • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads • A finite number of possible values – a discrete random variable • An example of a discrete random variable • The number of heads in tossing 3 coins (a finite number of possible values) • There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads • A finite number of possible values – a discrete random variable • This fits our general concept that discrete random variables are often “counts of …”

Chapter 6 – Section 1 • Other examples of discrete random variables • Other examples of discrete random variables • The possible rolls when rolling a pair of dice • A finite number of possible pairs, ranging from (1,1) to (6,6) • Other examples of discrete random variables • The possible rolls when rolling a pair of dice • A finite number of possible pairs, ranging from (1,1) to (6,6) • The number of pages in statistics textbooks • A countable number of possible values • Other examples of discrete random variables • The possible rolls when rolling a pair of dice • A finite number of possible pairs, ranging from (1,1) to (6,6) • The number of pages in statistics textbooks • A countable number of possible values • The number of visitors to the White House in a day • A countable number of possible values

Chapter 6 – Section 1 • A continuousrandomvariable is a random variable that has an infinite, and more than countable, number of values • The values are any number in an interval • A continuousrandomvariable is a random variable that has an infinite, and more than countable, number of values • The values are any number in an interval • Continuous random variables are designed to model continuous variables (see section 1.1) • Continuous random variables are often “measurements of …”

Chapter 6 – Section 1 • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit • The possible values (assuming that we can measure temperature to great accuracy) are in an interval • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit • The possible values (assuming that we can measure temperature to great accuracy) are in an interval • The interval may be something like (–20,110) • An example of a continuous random variable • The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit • The possible values (assuming that we can measure temperature to great accuracy) are in an interval • The interval may be something like (–20,110) • This fits our general concept that continuous random variables are often “measurements of …”

Chapter 6 – Section 1 • Other examples of continuous random variables • Other examples of continuous random variables • The height of a college student • A value in an interval between 3 and 8 feet • Other examples of continuous random variables • The height of a college student • A value in an interval between 3 and 8 feet • The length of a country and western song • A value in an interval between 1 and 15 minutes • Other examples of continuous random variables • The height of a college student • A value in an interval between 3 and 8 feet • The length of a country and western song • A value in an interval between 1 and 15 minutes • The number of bytes of storage used on a 80 GB (80 billion bytes) hard drive • Although this is discrete, it is more reasonable to model it as a continuous random variable between 0 and 80 GB

Chapter 6 – Section 1 • 1. A study was conducted to determine whether listening to classical music would increase one’s metabolism. To help determine the effect of classical music on the resting metabolic rate, the researchers measured several random variables on each of the subjects. For each of the variables, state whether it is discrete or continuous. Incidentally, the researchers found no significant effect of classical music on the resting metabolic rate. (Source: Carlsson E, Helgegren H and Slinde F (2005) Resting energy expenditure is not influenced by classical music. Journal of Negative Results in BioMedicine.) A. Discrete B. Continuous • a. Age at last birthday • b. Body mass index (body mass divided by the square of the height) • c. Resting energy expenditure (energy required to maintain basic bodily functions while resting for one-half hour) • d. Environmental temperature • e. Height (rounded to the nearest centimeter)

Chapter 6 – Section 1 • The probabilitydistribution of a discrete random variable X relates the values of X with their corresponding probabilities • The probabilitydistribution of a discrete random variable X relates the values of X with their corresponding probabilities • A distribution could be • In the form of a table • In the form of a graph • In the form of a mathematical formula

Chapter 6 – Section 1 • If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x • If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x • Examples • In tossing one coin, if X is the number of heads, then P(0) = 0.5 and P(1) = 0.5 • In rolling one die, if X is the number rolled, thenP(1) = 1/6

Chapter 6 – Section 1 • Properties of P(x) • Since P(x) form a probability distribution, they must satisfy the rules of probability • 0 ≤ P(x) ≤ 1 • ΣP(x) = 1 • In the second rule, the Σ sign means to add up the P(x)’s for all the possible x’s

Chapter 6 – Section 1 • An example of a discrete probability distribution • All of the P(x) values are positive and they add up to 1

Chapter 6 – Section 1 • An example that is not a probability distribution • Two things are wrong • An example that is not a probability distribution • Two things are wrong • P(5) is negative • An example that is not a probability distribution • Two things are wrong • P(5) is negative • The P(x)’s do not add up to 1

Chapter 6 – Section 1 • A probabilityhistogram is a histogram where • The horizontal axis corresponds to the possible values of X (i.e. the x’s) • The vertical axis corresponds to the probabilities for those values (i.e. the P(x)’s) • A probability histogram is very similar to a relative frequency histogram

Chapter 6 – Section 1 • An example of a probability histogram • The histogram is drawn so that the height of the bar is the probability of that value

Chapter 6 – Section 1 • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The values that have the higher probabilities are the ones that occur more often • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The values that have the higher probabilities are the ones that occur more often • The values that occur more often should have a larger role in calculating the mean • The meanofaprobabilitydistribution can be thought of in this way: • There are various possible values of a discrete random variable • The values that have the higher probabilities are the ones that occur more often • The values that occur more often should have a larger role in calculating the mean • The mean is the weighted average of the values, weighted by the probabilities

Chapter 6 – Section 1 • The mean of a discrete random variable is μX = Σ [ x • P(x) ] • The mean of a discrete random variable is μX = Σ [ x • P(x) ] • In this formula • x are the possible values of X • P(x) is the probability that x occurs • Σ means to add up these terms for all the possible values x

Multiply Multiply Multiply again Multiply again Multiply again Multiply again Multiply again Multiply again Chapter 6 – Section 1 • Example of a calculation for the mean • Example of a calculation for the mean • Example of a calculation for the mean • Example of a calculation for the mean • Add: 0.2 + 1.2 + 0.5 + 0.6 = 2.5 • The mean of this discrete random variable is 2.5

Chapter 6 – Section 1 • The calculation for this problem written out μX = Σ [ x • P(x) ] = [1• 0.2] + [2• 0.6] + [5• 0.1] + [6• 0.1] = 0.2 + 1.2 + 0.5 + 0.6 = 2.5 • The mean of this discrete random variable is 2.5

Chapter 6 – Section 1 • The mean can also be thought of this way (as in the Law of Large Numbers) • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • If we record the result each time • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • If we record the result each time • If we calculate the mean of the results (this is just a mean of a group of numbers) • The mean can also be thought of this way (as in the Law of Large Numbers) • If we repeat the experiment many times • If we record the result each time • If we calculate the mean of the results (this is just a mean of a group of numbers) • Then this mean of the results gets closer and closer to the mean of the random variable

Chapter 6 – Section 1 • The expectedvalue of a random variable is another term for its mean • The expectedvalue of a random variable is another term for its mean • The term “expected value” illustrates the long term nature of the experiments – as we perform more and more experiments, the mean of the results of those experiments gets closer to the “expected value” of the random variable

Chapter 6 – Section 1 • The variance of a discrete random variable is computed similarly as for the mean • The variance of a discrete random variable is computed similarly as for the mean • The mean is the weighted sum of the values μX = Σ [ x • P(x) ] • The variance of a discrete random variable is computed similarly as for the mean • The mean is the weighted sum of the values μX = Σ [ x • P(x) ] • The variance is the weighted sum of the squared differences from the mean σX2 = Σ [ (x – μX)2 • P(x) ] • The variance of a discrete random variable is computed similarly as for the mean • The mean is the weighted sum of the values μX = Σ [ x • P(x) ] • The variance is the weighted sum of the squared differences from the mean σX2 = Σ [ (x – μX)2 • P(x) ] • The standard deviation, as we’ve seen before, is the square root of the variance … σX = √ σX2

Chapter 6 – Section 1 • The variance formula σX2 = Σ [ (x – μX)2 • P(x) ] can involve calculations with many decimals or fractions • An equivalent formula is σX2 = [ Σx2 • P(x) ] – μX2 • This formula is often easier to compute

Chapter 6 – Section 1 • For variables and samples (section 3.2), we had the concept of a population variance (for the entire population) and a sample variance (for a sample from that population) • These probability distributions model the complete population • These are population variance formulas • There is no analogy for sample variance here

Chapter 6 – Section 1 • The variance can be calculated by hand, but the calculation is very tedious • Whenever possible, use technology (calculators, software programs, etc.) to calculate variances and standard deviations

Summary: Chapter 6 – Section 1 • Discrete random variables are measures of outcomes that have discrete values • Discrete random variables are specified by their probability distributions • The mean of a discrete random variable can be interpreted as the long term average of repeated independent experiments • The variance of a discrete random variable measures its dispersion from its mean

Example • Daniel Reisman, of Niverville, New York, submitted the following question to Marilyn vos Savant's December 27, 1998, Parade Magazine column, “Ask Marilyn:” • At a monthly ‘casino night,’ there is a game called Chuck-a-Luck: Three dice are rolled in a wire cage. You place a bet on any number from 1 to 6. If any one of the three dice comes up with your number, you win the amount of your bet. (You also get your original stake back.) If more than one die comes up with your number, you win the amount of your bet for each match. For example, if you had a $1 bet on number 5, and each of the dice came up with 5, you would win $3. • It appears that the odds of winning are 1 in 6 for each of the three dice, for a total of 3 out of 6 - or 50%. Adding the possibility of having more than one die come up with your number, the odds would seem to be in the gambler's favor. What are the odds of winning this game? I can't believe that a casino game would favor the gambler.

Example • Daniel computed the probabilities incorrectly. There are four possible outcomes. (The selected number can match 0, 1, 2, or 3 of the dice.) The random variable X represents the profit from a $1 bet in Chuck-A-Luck. The table below summarizes the probabilities of earning a profit of x dollars from a $1 bet. Use this table to answer the following questions. Number of dice matching the chosen number Profit, x Probability, P(X=x) • 0 dice $–1 125 / 216 • 1 dice $1 75 / 216 • 2 dice $2 15 / 216 • 3 dice $3 1 / 216

Example • a. Verify that this is a discrete probability distribution. • b. Draw a probability histogram. • c. Compute and interpret the mean of the random variable X. • d. Based on your answer to the previous question, would you recommend playing this game? • e. Compute the variance of the random variable X. • f. Compute the standard deviation of the random variable X. • g. What is the probability that a player will match all three of the dice? • h. What is the probability that a player will match at least one of the dice?

Example - Answers • a. (The probabilities are all between 0 and 1, and they sum to 1) • b. see Minitab • c. ($-0.08. This means that a gambler’s losses will be approximately $0.08 per game in the long run. If you play this game for a very long time, your mean winnings will be close to –0.0787 per game. So, in the long run, you are better off to throw away 7 cents than to play this game.) • d. (No) • e. (1.239) • f. (1.113) • g. (1/216) • h. (91/216)

Chapter 6 Section 1