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Class Note for MATH 1313 at UH


Class Note for MATH 1313 at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15
Math 1313 Section 83 Variance and Standard Deviation The mean or expected value of a random variable is important but it doesn t tell us the whole story about a probability distribution To get an accurate picture of the probability distribution we need more information We begin by looking at two histograms both of which have EX 3 03 025 02 015 El1 01 005 0 025 02 015 01 005 In the rst histogram the data are closely concentrated around the mean In the second histogram the data are widely dispersed A measure of the degree of dispersion or the spread of a probability distribution is called the variance A probability distribution with a small spread will have a smaller variance A probability distribution with a large spread will have a larger variance So comparing the two histograms we drew above the rst one will have a smaller variance than the second one We want to be able to compute the variance numerically Math 1313 Class Notes 7 Section 83 Page 1 of6 De nition Suppose a random variable X has the given probability distribution x x1 X2 X3 xquot PXx p1 p2 p3 pquot and has expected value E X u Then the variance of the random variable X is VarXp1x1 l2p2x22quot39pnxnlu2 Example 1 Find the variance of this probability distribution of a random variable x 1 0 2 4 PXx 15 32 29 24 Example 2 This example should help you understand the meaning of the variance Rose is considering investing 10000 in two mutual funds The anticipated returns from price appreciation and dividends in hundreds of dollars are described by the following probability distributions Mutual Fund A Mutual Fund B Returns Prob Returns Prob 4 02 2 02 8 05 6 04 10 03 8 04 a Compute the mean for the returns of each mutual fund Math 1313 Class Notes 7 Section 83 Page 2 0f6 b Which investment would provide Rose with the higher expected returns c Compute the variance for each probability distribution d Which investment would involve the lowest risk Variance is given in terms of the squares of the deviations from the mean The unit of measure of the variance is the square of the units in the data We often want to work with a measure given in the same units as the data so we will frequently want to work with the square root of the variance This quantity is called the standard deviation of the random variable Standard deviation is denoted by the Greek letter 039 Math 1313 Class Notes 7 Section 83 Page 3 of6 De nition The standard deviation of the random variable X is VarX where VarX is the variance of the random variable X Notation 039 VarX Example 3 The distribution of the number of raisins in a box of Raisin Bran is given in the following table Find the mean variance and standard deviation for the number of raisins in this cereal Raisins X 30 50 56 67 PXX 015 034 044 007 Math 1313 Class Notes 7 Section 83 Page 4 of6 Chebychev s Inequality We can use expected value and standard deviation to estimate some probabilities One type of problem is to determine the proportion of values of the random variable that lies within a given number of standard deviations of the mean These problems use Chebychev s Inequality We ll start by looking at what it means to be within k standard deviations of the mean Within 1 standard deviation of the mean Within 2 standard deviations of the mean Within 3 standard deviations of the mean Chebychev s Inequality LetX be a random variable with expected value u and standard deviation 039 Then the probability that a randomly chosen outcome of an experiment lies within k standard deviations of the expected value is at least 1 1 k2 In symbols we have Pu ko SX S uko Z 1 ki2 Math 1313 Class Notes 7 Section 83 Page 5 of6 Example 4 Suppose a random variable has a mean of 12 and a standard deviation of 4 Use Chebychev s Inequality to estimate the probability that an outcome of this experiment lies between 4 and 20 Example 5 A string of Christmas tree lights has an expected life of 200 hours and a standard deviation of 2 hours Estimate the probability that one of these strings of lights will last between 190 and 210 hours Math 1313 Class Notes 7 Section 83 Page 6 of6


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