1. 1. Variance of Probability Distribution 2. Spread 3. Standard Deviation 4. Unbiased Estimate 5. Sample Variance and Standard Deviation 6. Alternative Definitions 7. Chebychev's Inequality 1

2.  Let X be a random variable with values x 1, x 2,  …, x N and respective probabilities p 1, p 2,…, p N. The variance of the probability distribution is 2

3.  Roughly speaking, the variance measures the dispersal or spread of a distribution about its mean. The probability distribution whose histogram is drawn on the left has a smaller variance than that on the right. 3

4.  The standard deviation of probability distribution is 4

5.  Compute the variance and the standard deviation for the population of scores on a five- question quiz in the table. 5

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7.  If the average of a statistic, if that statistic were computed for each sample, equals the associated parameter for the population, then that statistic is said to be unbiased. 7

8.  The unbiased variance for a sample is  The unbiased standard deviation for a sample is 8

9.  Compute the sample variance and standard deviation for the weekly sales of car dealership A. 9

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11.  Two alternative definitions for variance are  and, for a binomial random variable with parameters n, p, and q, 11

12.  Find the variance when a fair coin is tossed 5 times and X is the number of heads. 12

13.  Chebychev's InequalitySuppose that a probability distribution with numerical outcomes has expected value and standard deviation Then the probability that a randomly chosen outcome lies between - c and + c is at least 13

14.  A drug company sells bottles containing 100 capsules of penicillin. Due to bottling procedure, not every bottle contains exactly 100 capsules. Assume that the average number of capsules in a bottle is 100 and the standard deviation is 2. If the company ships 5000 bottles, estimate the number of bottles having between 95 and 105 capsules, inclusive. 14

15.  The number of bottles containing between 100 - 5 and 100 + 5 capsules can be estimated using Chebychev's Inequality. 15 On average, at least 84% will contain between 95 and 105 capsules. 4200 bottles.

16.  The variance of a random variable is the sum of the products of the square of each outcome's distance from the expected value and the outcome's probability. The variance of the random variable X can also be computed as E( X 2 ) - [E( X )] 2.  A binomial random variable with parameters n and p has expected value np and variance np (1 - p ). 16

17.  The square root of the variance is called the standard deviation.  Chebychev's Inequality states that the probability that an outcome of an experiment is within c units of the mean is at least, where is the standard deviation. 17