7 5 Random variables probability distributions and expected

7. 5 Random variables, probability distributions, and expected value In this section, we will discuss concepts that are used in statistics.

Discrete random variable A variable that takes on different values according to chance. The variable includes only whole numbers ( no fractions or irrational numbers). Values of this variable include 0 , 1, 2, 3, 4, . . .

Discrete probability distribution • -consists of a correspondence between the values of the random variable and its associated probability • Properties: • The first property states that the probability of a random variable x must be a decimal number between 0 and 1 inclusive. • The second property states that the sum of all the individual probabilities must always equal one.

Example 1: • x=no. of customers in line waiting for a bank teller • x P(x) • 0. 07 • 1. 10 • 2. 18 • 3. 23 • 4. 32 • 5. 10 • Why is this a discrete probability distribution? • Answer: Variable x is discrete since it consists of whole numbers. The sum of the probabilities is one and all probabilities are between 0 and 1 inclusive.

Theoretical Discrete Probability Distribution • A bag consists of 2 black checkers and 3 red checkers. • Two checkers are drawn without replacement from this bag and the number of red checkers is noted. • Let x = number of red checkers drawn from this bag. • Determine the probability distribution of x and complete the table: • x P(x)

Probability distribution of x, the number of red checkers l l l Possible values of x are : 0, 1, 2 Why? P(x = 0 ) = P(black on first draw and black on second draw) = l l l Find P( x = 2) first, since its probability is easier to compute than P( x = 1) l Now, complete the rest of the table: x P(x) 0 1/10 1 2

Mean of probability distribution • To find the mean, we can use the familiar formula

Mean of probability distribution • x= number of heads • Recall, the sample space for tossing three coins: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT • x = 0, 1, 1, 1, 2, 2, 2, 3

Mean • Notice the outcomes of x = 1 and x = 2 occur three times each, while the outcomes x = 0 and x = 3 occur once each: The mean number of heads is the sum of the different outcomes divided by the total number of outcomes:

General Formula for Mean of Probability Distribution • Mean = where x is the value of the random variable and p(x) is the probability of occurrence for the random variable x.

Mean • x p(x) • • • 1/8 3/8 1/8 0 1 2 3 xp(x) 0(1/8)= 0 1(3/8)=3/8 2(3/8)=6/8 3(1/8)=3/8 sum = 3/2 = 1. 5 • Thus, mean number of heads = 1. 5 How is this number interpreted? Obviously, you cannot toss three coins and have an outcome of 1. 5 heads. The number 1. 5 is the “expected number” or long-range average. What this means is that if you performed this experiment thousands of times (tossing 3 coins), the average number of heads to appear would be 1. 5, exactly one-half of 3.

Application to Business • • A rock concert producer has scheduled an outdoor concert for Saturday, March 8. If it does not rain, the producer stands to make a $20, 000 profit from the concert. If it does rain, the producer will be forced to cancel the concert and will lose $12, 000 ( rock star’s fee, advertising costs, stadium rental, etc. ) The producer has learned from the National Weather Service that the probability of rain on March 8 is 0. 4. A) Write a probability distribution that represents the producer’s profit. b) Find the producer’s “expected profit”.

How much money should the producer expect to make in the long-range? • p(x) There are two possibilities: It x rains on March 8 or it doesn’t. Let x represent the random variable of money the producer will make. -12, 000 0. 4 So, x can either be $20, 000 (if it rains doesn’t rain) or x = $-12, 000 (if it does rain. Therefore, our table can be constructed at the right: does not 20, 000 0. 6 rain • The expected value is interpreted as the mean of the probability distribution (a long-range average). The number $7, 200 means that in the long –range, the producer would be ahead by$7, 200. It does not mean he will make exactly $7, 200 on March 8. He will either lose 12, 000 or gain 20, 000. x*p(x) -4, 800 12, 000 =7, 200