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# Class Note for MATH 1313 with Professor Flagg at UH

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Date Created: 02/06/15
Math 1313 Section 85 Normal Distribution In addition to these notes you will need to bring a copy of pages 1177 7 1178 the standard normal distribution table to class This table is located in an appendix in the online text In this section we will study normally distributed random variables These are continuous random variables We can see the difference between nite discrete random variables and continuous random variables by looking at graphs of them side by side 03 025 02 5 a 3 2 r r 2 3 a 015 01 005 With a nite discrete random variable we can create a table of values and list each number to which the random variable assigns a value With a continuous random variable we can t do that There are an in nite number of values We will look at a probability density function instead Here are properties of probability density functions 1 fx gt 0 for all values ofx 2 The area between the curve and the x aXis is 1 Some probability density functions have additional properties These are called normal distributions 3 there is a peak at the mean 4 the curve is symmetric about the mean 5 687 of the data is within one standard deviation of the mean 9545 of the area is within two standard deviations of the mean 9973 of the area is within three standard deviations of the mean Math 1313 Class Notes 7 Section 85 Page 1 of7 6 the curve approaches the x axis as x extends indefinitely in either direction Some normal distributions have additional properties These are called standard normal distributions 7 mean is 0 8 standard deviation is l Notation Pa SX SbPaltX bPaSX ltbPaltXltb sowe will use Pa lt X lt b as our standard notation This is true because the area under a point is 0 Notation We will denote the random variable which gives us the standard normal distribution by Z Math 1313 Class Notes 7 Section 85 Page 2 of 7 Using the Standard Normal Distribution Table to Find Probabilities The standard normal distribution mble gives the area between the curve and the x axis to the le of the line x z This area corresponds to the probability that Z is less than 2 or PZ lt z We ll use the table together with the properties listed above to answer many types of questions How to read the mble For a particular value 2 of our random variable 2 will be rounded to the nearest hundredth the probability in the table is PZltz This number corresponds to the area under the curve left ofthe vertical line x z Ifz 135 to find the probability PZlt135 from the table find the row with 13 in the rst column and then find the column headed by 005 The number that is in the 13 row and the 005 column is the probability that the random variable will have a value LESS than 135 We will use the basic rules ofprobability and the table to do the exercises There are several ditferent techniques to using the table and we will walk through them one at a time 1 Probability Less than a number Example 1 Find PZlt136 Math 1313 Class Notes 7 Section 85 Page 3 of7 Example 2 Find PZ lt 047 11 Probability Greater than a number This is the area under the curve from the vertical line X z to the right The total area under the curve is 1 so 1 7 area to the left ofX 2 equals area to the right ofz OR since the graph of the normal distribution is symmetric with respect to the y aXis the area to the RIGHT of X z is the SAME as the area to the LEFT of X 2 So to nd PZ gt 2 look up z in the chart and that is the probability EXample 3 Find PZ gt 178 EXample 4 Find PZ gt 105 111 Probability Between two numbers The probability that the random variable takes on a value between two numbers is the area under the curve between the vertical lines at these values This area is equivalent to the area to the left of the larger number minus the area to the left of the smaller number So to find the probability between two numbers look up both numbers on the chart and take the difference PaltZltbPZltb7PZlta Example 5 Find P l25 lt Z lt 203 Math 1313 Class Notes 7 Section 85 Page 4 of7 Example 6 Find P 68 lt Z lt 141 Sometimes we want the nd 2 the number inside the parentheses given a probability IV Find z if PZ lt z a palticular number Look up the number in the chart The row heading plus the column heading is 2 Example 7 Find 2 if PZ lt z 8944 Example 8 Find 2 if PZ lt z 0401 V Find z if PZ gt z a palticular number This says that the area to the right of the number 2 is the number given So the area to the LEFT of 72 is this probability Look up the probability on the chart nd the row and column The answer is the NEGATIVE of this number Example 9 Find 2 if PZ gt z 9463 Example 10 Find 2 if PZ gt z 0132 There is also another version of this Example 105 Find 2 if PZlt z 3228 Math 1313 Class Notes 7 Section 85 Page 5 of7 VI Find z if the probability between z and z is given We are given a probability which is the area under the curve from 72 to z for some number 2 How to nd 2 Take the number given add 1 then divide that by 2 Look this number up on the chart the column plus row is 2 Example 11 Findz if P z lt Z lt z 9812 Example 12 Find 2 if P z lt Z lt z 5408 Example 13 Find 2 if P z lt Z lt z 1820 Next we need to look at what to do with a distribution that is normal but not standard normal that is a normally distributed random variable with mean other than 0 and standard deviation other than 1 Math 1313 Class Notes 7 Section 85 Page 6 of7 We can convert any problem involving probability of a normally distributed random variable to one with a standard normal random variable thus allowing us to use the table Here s how Suppose X is a normal random variable with E X u and standard deviation 039 ThenX can be converted to the standard normal random variable using the formula Then we can evaluate the new problem using the techniques presented earlier in the lesson Example 14 Suppose X is a normally distributed random variable with u 50 and 039 30 Find PX lt 95 Example 15 Suppose X is a normally distributed random variable with u 85 and 039 16 Find PX gt 54 Example 16 Suppose X is a normally distributed random variable with u 100 and 039 20 Find P85 lt X lt 110 Math 1313 Class Notes 7 Section 85 Page 7 of7

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