Download presentation
Presentation is loading. Please wait.
Published byKatherine Williams Modified over 8 years ago
1
1 Chapter 16 Random Variables Random Variables and Expected Value
2
2 Betting on Death! Many people in America have life insurance policies. Although you might not want to think of it this way, when you purchase a life insurance policy, you’re betting that you will die sooner rather than later… Although this is a bet that people really don’t want to win, it is a bet that they are willing to take just to be sure that their families are financially secure in the event of death. Most families depend on the income of one or more people in the household. What would happen if that income suddenly disappeared? Life insurance help us handle such disasters. When you purchase a life insurance policy, it’s in your best interests that the company makes a profit and does well; why do you think that is?
3
3 Betting on Death! Question: You purchase a policy that charges only $50 a year. If it pays $10,000 for death and $5000 for a permanent disability, is the company likely to make a profit? Actuaries at for the company have determined the following probabilities in any given year: P (Death) = 1/1000 P (Permanently disabled) = 2/1000 P (Healthy) = 997/1000 We’ll come back to this problem later on…
4
4 Random Variables and Expected Value Random Variable A Random Variable is a variable whose values are numbers that are determined by an outcome of a random event. values Random variables are denoted by capital letters, while the values of random variables are denoted with lowercase letters (small letters) mean expected value The mean of the discrete random variable, X, is also called the expected value of X. Notationally, the expected value of X is denoted by E(X). It is what we expect to happen. The formula for expected value is:
5
5 Examples In the experiment of flipping three coins, consider the outcomes and define the random variable X as the number of heads that appear. The outcomes are {no heads, 1 head, 2 heads, or 3 heads} X has values in the set: {0, 1, 2, 3 When rolling two dice and finding the sum, determine the outcomes and the random variable Y. The outcomes are {(1,1), (1,2), (1,3), (1,4), (1,5), etc…} Y has values in the set: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} In our life insurance example, what are the outcomes and random variables Z if we define them as the possible payments. The outcomes are {die, disabled, healthy} Z has values in the set: {$10,000, $5000, $0}
6
6 Back to Betting on Death So, will the company make a profit for any given year? How much will they make or lose? These questions are answered by finding the expected value. The expected Value is: Policyholder Outcome PayoutxProbability P(X = x) Die $10,0001/1000 Disability $50002/1000 Healthy $0997/1000
7
7 Back to Betting on Death So, what does this mean? Since each customer pays $50 per year, the company expects to make a profit of $30 per customer per year. The expected value for the company is a payout, on average, of $20 per customer per year. Since each customer pays $50 per year, the company expects to make a profit of $30 per customer per year. expected average payout It’s important to note that the insurance company will never really pay anyone $20; it only pays $10,000, $5000, or $0. $20 is the expected average payout given a large number of policy holders based on the LLN.
8
8 Labor Costs A car’s air conditioner recently needed to be repaired at the auto shop. The mechanic said that it could for $60 in 75% of the cases by drawing down and recharging the coolant. If that fails, it will cost an additional $140 to replace the unit. What are the outcomes, random variables, and the probability distribution? OutcomeCostxProbability P(X = x) Quick fix works$60¾ =.75 Replace unit$200¼ =.25
9
9 Labor Costs A car’s air conditioner recently needed to be repaired at the auto shop. The mechanic said that it could for $60 in 75% of the cases by drawing down and recharging the coolant. If that fails, it will cost an additional $140 to replace the unit. What is the expected value of the cost of this repair?
10
10 Labor Costs A car’s air conditioner recently needed to be repaired at the auto shop. The mechanic said that it could for $60 in 75% of the cases by drawing down and recharging the coolant. If that fails, it will cost an additional $140 to replace the unit. What does this mean in context of this problem? Car owners with this problem will spend an average of $95 to get their car fixed at this auto shop.
11
11 Got to Love Those Aces It takes $5 to play a game From a standard 52 card deck of cards, if you get an ace of hearts, you get $100 If you get any other ace, you get $10 If you get any other heart, you get your $5 back If you get any other card, you lose What is the expect value of this game and is it worth it to play this game?
12
12 Got to Love Those Aces Outcome X = Payout Probability: P(X = x) Ace of Hearts$951/52 =.0192 Other Aces$53/52 =.0577 Other Hearts$012/52 =.2308 Other Cards-$536/52 =.6923 First, you want to determine your possible winnings (let’s include the $5 cost) and probabilities: Now, we can find the expected value, E(X): Is this game worth playing?
13
13 Probability Histogram We can use histograms to display probability distributions as well as distributions of data.
14
14 Discrete Versus Continuous Earlier this year, we covered the notion of discrete and continuous, but it needs more exploration now… A discrete random variable has a countable number of outcomes. In other words, it is possible for you to count and make a list of all of the possible outcomes. Discrete random variables take on only integer values. Suppose, for example, that we flip a coin and count the number of heads. The number of heads results from a random process - flipping a coin. And the number of heads is represented by an integer value.
15
15 Continuous Random Variable Continuous random variables average Continuous random variables, in contrast, can take on any value within a range of values. For example, suppose we flip a coin many times and compute the average number of heads per flip. The average number of heads per flip results from a random process - flipping a coin. And the average number of heads per flip can take on any value between 0 and 1, even a non-integer value. Therefore, the average number of heads per flip is a continuous random variable. A continuous random variables are not countable. In other words, you cannot list every single possible outcome. For example, the amount of water can you put into a 5-gallon container – there are an infinite number of possibilities.
16
16 Example Which of the following is a discrete random variable? I. The average height of a randomly selected group of boys. II. The annual number of sweepstakes winners from New York City. III. The number of presidential elections in the 20th century. (A) I only (B) II only (C) III only (D) I and II (E) II and III
17
17 Solution The correct answer is B. The annual number of sweepstakes winners is an integer value and it results from a random process; so it is a discrete random variable. The average height of a group of boys could be a non-integer, so it is not a discrete variable. And the number of presidential elections in the 20th century is an integer, but it does not vary and it does not result from a random process; so it is not a random variable.
18
18 Assignment Chapter 16 Lesson : Random Variables and Expected Value Read : Chapter 16 Problems : 1 – 41 (odd)
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.