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3.slide

* In other words, what is unpredictable and chancy in case of an individual is predictable and uniform in the case of a large group. * This law forms the basis for the expectation of probable-loss upon which insurance premium rates are computed. Also called law of averages.

Law of Large Numbers

Observe a random variable X very many times. In the long run, the proportion of outcomes taking any value gets close to the probability of that value. The Law of Large Numbers says that the average of the observed values gets close to the mean μ X of X.

4.slide ; Law of Large Numbers for Discrete Random Variables * The Law of Large Numbers, which is a theorem proved about the mathematical model of probability, shows that this model is consistent with the frequency interpretation of probability.

5.slide ; Chebyshev Inequality * To discuss the Law of Large Numbers, we first need an important inequality called the Chebyshev Inequality. * Chebyshev’s Inequality is a formula in probability theory that relates to the distribution of numbers in a set. * This formula is able to prove with little provided information the probability of outliers existing at a certain interval.

6.slide * Given X is a random variable, A stands for the mean of the set, K is the number of standard deviations, and Y is the value of the standard deviation, the formula reads as follows: * Pr(|X-A|=>KY)<=1/K2, Theabsolute value of the difference of X minus A is greater than or equal to the K times Y has the probability of less than or equal to one divided by K squared.

7.slide Statement of Chebyshev’s Inequality * Chebyshev’s inequality says that at least 1-1/K2 of data from a sample must fall within K standard deviations from the mean, where K is any positivereal number greater than one. We can also state the inequality above by replacing the phrase “data from a sample” with probability distribution. This is because Chebyshev’s inequality is a result from probability, which can then be applied to statistics.

8.slide Illustration of the Inequality

To illustrate the inequality, we will look at it for a few values of K: * For K = 2 we have 1 – 1/K2 = 1 - 1/4 = 3/4 = 75%. So Chebyshev’s inequality says that at least 75% of the data values of any distribution must be within two standard deviations of the mean. * For K = 3 we have 1 – 1/K2 = 1 - 1/9 = 8/9 = 89%. So Chebyshev’s inequality says that at least 89% of the data values of any distribution must be within three standard deviations of the mean.

9.slide The Long Run and the Expected Value

Random experiments and random variables have long-term regularities. The Law of Large Numbers says that in repeated independent trials, the relative frequency of each outcome of a random experiment tends to approach the probability of that outcome. That implies that the long-term average value of a discrete random variable in repeated experiments tends to approach a limit, called the expected value of the random variable.

10.slide

The expected value of a random variable depends only on the probability distribution of the random variable. The expected value has properties that can be exploited to find the expected value of some complicated random variables in terms of simpler ones. These properties allow us to find the expected value of the sample sum and sample mean of random draws with and without replacement from a box of numbered tickets.

11.slide :The Law of Large Numbers says that in repeated independent trials with probability p of success in each trial, the chance that the fraction of successes is close to p grows as the number of trials grows. More precisely, for any tolerance e>0,

P(| (fraction of successes in n trials) - p | < e) approaches 100% as the number n of trials grows. This expresses a long-term regularity of repeated independent trials with a shared probability of success.

12.slide: The Expected Value of the Sample Sum of n random Draws from a Box

If a box contains tickets labeled with numbers, the expected value of the sample sum of the labels on n tickets drawn at random with or without replacement from the box is n×(average of the labels on the tickets in the box), where (average of the labels on the tickets in the box) means the average of the list of numbers on all the labels, including repeated values as many times as they occur on different tickets.

(If the draws are without replacement, the number of draws cannot exceed the number of tickets in the box.)

13.slide Exercise A box contains 12 tickets labeled with numbers. The number on the tickets are

-9, -7, -6, -5, -4, -4, -3, -1, -1, 0, 5, 10 .

The expected value of the sample sum of the ticket labels in 11 independent random draws with replacement from the box is ??

[-Solution]

The expected value of the sum of the numbers on the tickets in n draws with replacement from a box of numbered tickets is n×(average of the numbers on the tickets). The average of the numbers on the tickets in this box is -2.08 and the number n of draws is 11, so the expected value is

(11)×(-2.08) = -22.92.

14.slide:Expected Value of the Sample Mean and Sample Percentage

The sample mean is the sample sum divided by the sample size, n. Dividing by the sample size is the same as multiplying by its reciprocal, 1/n, so we can use the properties of expectation to find the expected value of the sample mean from the expected value of the sample sum: The expected value of the sample mean is the expected value of the sample sum, times 1/n. Because the expected value of the sample sum is n×(average of the numbers on all the tickets in box), whether the sampling is with or without replacement, the expected value of the sample mean is the average of the numbers on all the tickets in the box, for random sampling with or without replacement.

15.slide Summary * In repeated independent trials with the same probability of success, as the number of trials increases, the fraction of successes is increasingly likely to be close to the probability of success in each trial. This is the Law of Large Numbers. * As a consequence of the Law of Large Numbers, if a discrete random variable is observed repeatedly in independent experiments, the fraction of experiments in which the random variable equals any of its possible values is increasingly likely to be close to the probability that the random variable equals that value. * The mean of the observed values of the random variable in repeated independent experiments is thus increasingly likely to be close to a weighted average of the possible values, where the weights are the probabilities of the values * That weighted average is called the expected value of the random variable. The expected value of a random variable depends only on the probability distribution of the random variable, so we can speak interchangeably of the expected value of a random variable or of its probability distribution.…...

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