1. Population Versus Sample 2. Statistic Versus Parameter 3. Mean (Average) of a Sample 4. Mean (Average) of a Population 5. Expected Value 6. Expected Value of Binomial Trial 1

A population is a set of all elements about which information is desired. A sample is a subset of a population that is analyzed in an attempt to estimate certain properties of the entire population. 2

 A clothing manufacturer wants to know what style of jeans teens between 13 and 16 will buy. To help answer this question, 200 teens between 13 and 16 were surveyed.  The population is all teens between 13 and 16.  The sample is the 200 teens between 13 and 16 surveyed. 3

 A numerical descriptive measurement made on a sample is called a statistic. Such a measurement made on a population is called a parameter of the population.  Since we cannot usually have access to entire populations, we rely on our experimental results to obtain statistics, and we attempt to use the statistics to estimate the parameters of the population. 4

 Let an experiment have as outcomes the numbers x 1, x 2, …, x r with frequencies f 1, f 2,…, f r, respectively, so that f 1 + f 2 +…+ f r = n. Then the sample mean equals  or 5

 If the population has x 1, x 2,…, x r with frequencies f 1, f 2,…, f r, respectively. Then the population mean equals  or 6 Note: Greek letters are used for parameters.

 An ecologist observes the life expectancy of a certain species of deer held in captivity. The table shows the data observed on a population of 1000 deer. What is the mean life expectancy of this population? 7

 The relative frequencies are given in the table. 8

 The expected value of the random variable X which can take on the values x 1, x 2,…,x N with  Pr( X = x 1 ) = p 1, Pr( X = x 2 ) = p 2,…, Pr( X = x N ) = p N  is  E( X ) = x 1 p 1 + x 2 p 2 + …+ x N p N. 9

 The expected value of the random variable X is also called the mean of the probability distribution of X and is also designated by  The expected value of a random variable is the center of the probability distribution in the sense that it is the balance point of the histogram. 10

 Five coins are tossed and the number of heads observed. Find the expected value. 11

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 X is a binomial random variable with parameters n and p, then  E( X ) = np. 13

 Five coins are tossed and the number of heads observed. Find the expected value.  A "success" is a head and p =.5.  The number of trials is n = 5.  E( X ) = np = 5(.5) =

 The expected value of a completely fair game is zero. 15

 Two people play a dice game. A single die is thrown. If the outcome is 1 or 2, then A pays B $2. If the outcome is 3, 4, 5, or 6, then B pays A $4. What are the long-run expected winnings for A?  X represents the payoff to A. Therefore, X is either -2 or 4. 16

 Pr( X = -2) = 2/6 = 1/3  Pr( X = 4) = 4/6 = 2/3  E( X ) = -2(1/3) + 4(2/3) = 2  On average, A should expect to win $2 per play.  If A paid B $4 on 1 and 2 but B paid A $2 on 3, 4, 5, and 6, then E( X ) = 0 and the game would be fair. 17

 The sample mean of a sample of n numbers is the sum of the numbers divided by n.  The expected value of a random variable is the sum of the products of each outcome and its probability. 18