Section 5.1 Expected Value HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

Random variable – a variable whose numeric value is determined by the outcome of a random experiment. Probability distribution – a table or formula that lists the probabilities for each outcome of the random variable, X. Discrete random variable – a variable that may take on either finitely many values, or have infinitely many values that are determined by a counting process. Discrete probability distribution – a table or formula that lists the probabilities for each outcome of the discrete random variable, x. HAWKES LEARNING SYSTEMS math courseware specialists Definitions: Probability Distribution 5.1 Expected Value

Create a probability distribution for X, the sum of two rolled dice. Create the probability distribution: HAWKES LEARNING SYSTEMS math courseware specialists Solution: To begin, list all possible values of X. Then, to find the probability distribution, we need to calculate the probability of each outcome. Probability Distribution 5.1 Expected Value Rolling Two Dice x P(X = x)

Expected Value: HAWKES LEARNING SYSTEMS math courseware specialists The expected value, E(X), for a discrete probability distribution is the mean of a probability distribution Probability Distribution 5.1 Expected Value Formula:

HAWKES LEARNING SYSTEMS math courseware specialists Probability Distribution 5.1 Expected Value You are trying to decide between two different investments options. The two plans are summarized in the table below. The left-hand column for each plan gives the potential profits, and the right-hand columns give their respective probabilities. Which plan should you choose? Determine the expected value: Investment AInvestment B $1200P = 0.1$1500P = 0.3 $950P = 0.2$800P = 0.1 $130P = 0.4–$100P = 0.2 –$575P = 0.1–$250P = 0.2 –$1400P = 0.2–$690P = 0.2

HAWKES LEARNING SYSTEMS math courseware specialists Probability Distribution 5.1 Expected Value Solution: It is difficult to determine which plan is better by simply looking at the table. Let’s use the expected value to compare the plans. For Investment A: For Investment B : E(X)  (1200)(0.1)  (950)(0.2)  (130)(0.4)  (–575)(0.1)  (–1400)(0.2)  120  190  52   280  E(X)  (1500)(0.3)  (800)(0.1)  (–100)(0.2)  (–250)(0.2)  (–690)(0.2)  450  80  20  50  138  322 Best option

Variance for a Discrete Probability Distribution: HAWKES LEARNING SYSTEMS math courseware specialists Probability Distribution 5.1 Expected Value Standard Deviation for a Discrete Probability Distribution: Round the variance and standard deviation to one more decimal place than what is given in the data set.

Which of the investment plans in the previous example carries more risk, Plan A or Plan B? Determine the risk: HAWKES LEARNING SYSTEMS math courseware specialists Solution: To determine which plan carries more risk, we need to look at their variances. For Investment A: Probability Distribution 5.1 Expected Value xP(X = x)x ∙ P(X = x)x 2 ∙ P(X = x) $1200P = ,000 $950P = ,500 $130P = –$575P = 0.1–57.533,062.5 –$1400P = 0.2–280392,000

HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Using the equation to determine the variance for a discrete probability distribution we have For Investment B: Probability Distribution 5.1 Expected Value xP(X = x)x ∙ P(X = x)x 2 ∙ P(X = x) $1500P = ,000 $800P = ,000 –$100P = 0.2– –$250P = 0.2–5012,500 –$690P = 0.2–13895,220  755,

HAWKES LEARNING SYSTEMS math courseware specialists Solution (continued): Using the equation to determine the variance for a discrete probability distribution we have Since the variance of profits of Plan B is slightly less than those in Plan A we can conclude that Plan B carries a slightly lower amount of risk. Probability Distribution 5.1 Expected Value  745,036.