1. chapter5 Statistical and probabilistic concepts, Implementation to Insurance Subjects of the Unit 1.Counting 2.Probability concepts 3.Random Variables 4.Used distributions In Insurance 1

2. 1. Counting Since many (not all) probability problems will be solved by counting outcomes, we will develop a number of counting principles which will prove useful in solving probability problems. In counting, we will use the convenient notation :the number of elements in the set (or event) A. Example 1: A neighborhood association has 100 families on its membership list. 78 of the families have a credit card and 50 of the families are currently paying off a car loan. 41 of the families have both a credit card and a car loan. A financial planner intends to call on one of the 100 families today. The planner's sample space consists of the families in the association. The events of interest to the planner are the following:

3. Counting (2) C: the family has a credit card L: the family has a car loan We are given the following information: The planner is also interested in the answers to some other questions For example, she would first like to know how many families do not have credit cards. Since there are 100 families and 78 have credit cards, the number of families that do not have credit cards is: 100 – 78 = 22 This can be written using the counting notation as

4. Rules and B For any finite sample space S and events A

5. A combination A combination of n objects taken r at a lime is an r-element subset of the original n elements (or, equivalently, an unordered selection of r of the original n elements). Notation: The number of combinations of n elements taken r at a time is denoted by. Counting Principle for Combination

6. Example 2. A company has ten management trainees. The company will test a new training method on four of the ten trainees. In how many ways can four trainees be selected for testing? Solution.

7. 2. Probability concepts Approach 1. Equally likely outcomes Approach 2. Relative frequency

8. Approach 1. Equally likely outcomes Probability by Counting for equally likely outcomes : Example 3: Suppose you are rolling a single six-sides die whose sides bear the numbers 1,2,3,4,5 and 6. You wish to bet on the event that you will roll a number less than 4. since the outcomes 1,2,3 are less than 4. The probability of this event is 3/6,

9. Definition. Let be an event from a sample space in which all outcomes are equally likely. The probability of denoted, is defined by Example 4. A company has 200 employees. 50 of these employees are smokers. One employee is selected at random. What is the probability that the selected employee is a smoker (Sm)? Solution

10. Approach 2. Relative frequency Relative Frequency estimate of the probability of an event Example. Assume that you are tossing a coin and suspect that it is not fair. Then the probability of tossing a head can not be determined by counting, we have to estimate the probability by tossing the coin a large number of times and count the number of heads. If you toss the coin 1000 times and observe 750 heads, your best estimate of the probability of a head of one toss is 750/1000=0.75

11. The language of Probability Definition 1. A set is a collection of objects such as the numbers 1,2,3,4,5 and 6. These objects are called the elements or members of the set. If the set is finite and small enough that we can easily list all of its elements, we can describe the set by listing all of its elements in braces. Example 1. The set of all positive real numbers may be written as: Definition 2. The sample space,S for a probability experiment is the set of all possible outcomes of experiment. Example 2. A coin is tossed and the side facing up is recorded. The sample space is :

12. A set Example 3. (Death of an insured) An insurance company is interested in the probability that an insured will die in the next year. The sample space is: Example 4. An insurance company has sold 100 individual life insurance policies. When an insured individual dies, the beneficiary named in the policy will file a claim for the amount of the policy. You wish to observe the number of claims filed in the next year. The sample space consists of all integers from 0 to 100, so,

13. An event Example 5. An insurance company sells life insurance to a 30-year-old female. The company is interested in the age of the insured when she eventually dies. If the company assumes that the insured will not live to 110, the sample space is Definition 3. An event is a subset of the sample space S. Example 6. A coin is tossed. You wish to find the probability of the event "toss a head." The sample space is The event is the subset Example 7. An insurance company has sold 100 individual life policies. The company is interested in the probability that at most 5 of the policies have death benefit claims in the next year. The sample space is The event is the subset

14. Negation: The event not E is written as The Compound Events A or B: Example: Rolling a single die. Suppose that we have the opportunity to bet on two different events: A: an even number is rolled A={2,4,6} B: a number less than 5 is rolled B={1,2,3,4} If we bet that A or B occurs, we will win if any element of the two sets above is rolled. A or B={1,2,3,4,6 } In forming the set for A or B we have combined the sets A and B by listing all outcomes which appear in either A or B. The resulting set is called the union of A and B, and is written as: A U B. It should be clear that for any two events A and B;

15. The Compound Events A and B: When one decide to bet on the event A and B. In that case, both the event A and the event B must occur on the single roll. This can happen only if an outcome occurs which is common to both events. A and B={2,4} In forming the set for A and B we have listed all outcomes which are in both sets simultaneously. This set is referred to as the intersection of A and B, and is written as. :For any two events A and B

16. Probability rules Definition: two events are mutually exclusive if Addition rule for mutually exclusive event:

17. Conditional probability 1. Conditional probability by counting Example. A health insurance pool includes 200 individuals. The insurer is interested in the number of smokers in the pool among both males and females. The following table (called a contingency table) shows the desired numbers. TotalFemales (F)Males (M) 502228Smokers 1507872Non Smokers 200100 Total

18. Suppose one individual is to be chosen at random. Counting can be used to find the probability that the individual is a male, a female, a smoker, or both. Suppose you were told that the selected individual was a male, and asked for the probability that the individual was a smoker, given that the individual was a male.

19. Definition For any two events A and B, the conditional probability of A given B is defined as follows: Example. An automobile insurance company does a study to find the probability for the number of claims that a policyholder will file in a year. Their study gives the following probabilities for the individual outcomes 0,1,2,3. 3210Number of claims 0.010.050.220.72Probability Find the probability that a policyholder files exactly 2 claims, given that the policyholder has filed at least one claim.

20. Solution. Let C be the event that at least one claim is filed. Then and We also need the value Then This tells us that approximately 17.9% of the policyholders who file claims will file exactly 2 claims.

21. 3. Random Variables Definition. A random variable is a numerical quantity Variable whose value depends on chance. Example. You are tossing a coin twice and will bet on the number of heads. The outcome is a number (0, 1 or 2) which depends on chance. The number of heads is a random variable.

22. The Probability Function of a Discrete Random variable Definition. Let be a discrete random variable. A probability function for is a function which assigns a probability to each value of the random variable. Such that The probability function is also referred to as the probability mass function or the discrete density function for. Example. an automobile insurer studied the number of claims filed by a policyholder in a given year. The probability function was as follows. 3210Number of claims 0.010.050.220.72

23. The cumulative distribution Definition. Let be a random variable. The cumulative distribution function for is defined by Example. The cumulative distribution function for the previous example is given by the following table: 3210Number of claims 10.990.940.72F(X) This tells us that 94% of policyholders file one claim or less in a year leaving 6% who file more than one claim

24. 4. Used distributions In Insurance Discrete distributions -Binomial distribution -Poisson distribution Continuous distributions -Exponential distribution

25. a. Binomial distribution If X is a binomial random variable with n trials and, The mean and variance of binomial distribution 10Number of success (x) pq= 1-pP(x) The VarianceThe mean

26. Example The probability of car accident within one year for novice policyholders equals to 0.2. Assume that you are an insurance agent with 10 novice clients. You are willing to suppose that car accidents are independent events. 1. What is the probability of no accident in the next year? 2. What is the probability that that one accident will occur? 3. Calculate the mean and the variance of non accident.

27. Solution If accidents are independent, the number of non accident X will be a binomial random variable with parameter n=10 and 3. The mean and the variance of non accident

28. b. Poisson Distribution The random variable X follows the Poisson distribution with parameter (or average rate) ) if The mean and variance of binomial distribution The VarianceThe mean

29. c. Exponential distribution The random variable X follows an exponential distribution with parameter if The cumulative and survival function The mean and the variance are: The VarianceThe mean SurvivalCumulative

30. Example An insurance company is studying the reliability of a system. The time (in hours) from installation to failure of the system is a random variable. The study shows that T follows an exponential distribution with. 1. What is the probability that the system fails within 100 hours? 2. Find the probability that the part lasts for more than 500 hours. 3. Calculate the mean and the variance of the time failure of the system.

31. Solution 1.The probability that a part fails within 100 hours is 2.The probability that the part lasts for more than 500 hours 3.The mean and the variance of the time failure of the system. Note. The system has an expected life of 1000 hours, you might not want to use it for 1000 hours if your life depended on it.