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Class Note for MATH 1313 at UH

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Date Created: 02/06/15
Section 83 Variance and Standard Deviation Section 83 Variance and Standard Deviation The Variance ofa random variable X is the measure of degree of dispersion or spread ofa probability distribution about its mean ie how much on average each of the values ofX deviates from the mean Note A probability distribution with a small large spread about its mean will have a small large variance Variance ofa Random Variable X Suppose a random variable has the probability distribution xx1x2xn PXx 701 702 701 and expected value EX u Then the variance of the random variable X is VarX 701061 m2 702062 m2 mm W Note We square each since some may be negative Standard Deviation measures the same thing as the variance The standard deviation ofa Random Variable X is 0 Warm 1061 mz P2x2 W 701061 mz Example 1 Compute the mean variance and standard deviation of the random variable X with probability distribution as follows X PXX 03 03 Section 83 Variance and Standard Deviation Example 2 An investor is considering two business ventures The anticipated returns in thousands of dollars of each venture are described by the following probability distributions Venture 1 Venture 2 Earnings Probability Earning Probability 5 02 15 015 30 06 50 075 60 02 100 010 a Compute the mean and variance for each venture b Which investment would provide the investor with the higher expected return the greater mean c Which investment would the element of risk be less that is which probability distribution has the smaller variance Section 83 Variance and Standard Deviation Chebychev39s Inequality LetX be a random variable with expected value u and standard deviation 0 Then the probability that a randomly chosen outcome of the experiment lies between u kaandy k0 is atleast 1 1 1 k 2that1sPu k0SXSyka21 Let s see What this is saying Example 3 A probability distribution has a mean 20 and a standard deviation of3 Use Chebychev s Inequality to estimate the probability that an outcome of the experiment lies between 12 and 28 Section 83 Variance and Standard Deviation Example 4 A light bulb has an expected life of 200 hours and a standard deviation of2 hours Use ChebycheV s Inequality to estimate the probability that one of these light bulbs will last between 190 and 210 hours Section 84 The Binomial Distribution Section 84 The Binomial Distribution A binomial experiment has the following properties 1 Number of trials is fixed 2 There are 2 outcomes of the experiment Success probability denoted by p and failure probability denoted by q Note p q 1 3 The probability of success in each trial is the same 4 The trials are independent of each other Experiments with two outcomes are called Bernoulli trials or Binomial trials Finding the Probability of an Event of a Binomial Experiment In a binomial experiment in which the probability of success in any trial is p the probability of exactlyX successes in n independent trials is given by PX x Cn xpxqquot x X is called a binomial random variable and its probability distribution is called a binomial probability distribution Example 1 on page 505 derives this formula Example 1 Consider the following binomial experiment A fair die is cast four times Compute the probability of obtaining exactly one 6 in the four throws Section 84 The Binomial Distribution Example 2 Let the random variable X denote the number ofgirls in a fivechild family If the probability of a female birth is 05 construct the binomial distribution associated with this experiment What is the probability of having at least one girl Mean Variance and Standard Deviation of a Random Variable le is a binomial random variable associated with a binomial experiment consisting ofn trials with probability of success p and probability of failure q then the mean EX variance and standard deviation ofX are given by applying the following formulas EX np VarX npq a VarX 1lnpq Section 84 The Binomial Distribution Example 3 Consider the following binomial experiment If the probability that a marriage will end in divorce within 20 years after its start is 06 what is the probability that out of 6 couples just married in the next 20 years all will be divorced a None will be divorced b Exactly two couples will be divorced c At least two couples will be divorced d At most five couples will be divorce Example 4 A baseball pitcher gives up a hit on the average of once every fifth pitch lf nine pitches are thrown what is the probability that a Exactly three of them result in a hit b None of them result in a hit Section 84 The Binomial Distribution Example 5 The probability ofa person contracting in uenza on exposure is 62 In the binomial experiment for a family of 12 that has been exposed What is the a mean b standard deviation c variance Section 85 The Normal Distribution Section 85 The Normal Distribution A random variable that may take on in nitely many values is called a continuous random variable The probability distribution associated with this type of random variable is called a continuous probability distribution A continuous probability distribution is defined by a function fcalled the probability density function The function has domain equal to those values the continuous random variable assumes The probability density function has the following properties 1 fx gt0 for all values ofX 2 The area between the curve and the X axis is 1 The probability that the random variable X associated with a given probability density function assumes a value in an interval a ltx lt b is given by the area of the region between the graph off and the Xaxis from X a to X b Here is a picture This value is Pa lt X lt b Note Pa lt X lt b Pa lt X lt b Pa lt X lt b Pa lt X lt b since the area under one pointis 0 Section 85 The Normal Distribution Normal Distributions For these types of distributions 1 The graph is a bellshaped curve 2 u and a each have the same meaning mean and standard deviation 3 u determines the location of the center of the curve 4 0 determines the sharpness or atness of the curve Also the normal curve has the following characteristics 1 The curve has peak atX u 2 The curve is symmetric with respect to the vertical line X u 3 The curve always lies above the X aXis but approaches the X aXis as X extends inde nitely in either direction 4 The area under the curve is 1 5 For any normal curve 6827 of the area under the curve lies within 1 standard deviation of the mean ie between u a and u a 9545 ofthe area lies within 2 standard deviations of the mean and 9973 of the area lies within 3 standard deviations of the mean M Since any normal curve can be transformed into any other normal curve we will study from here on the Standard Normal Curve The Standard Normal Curve has u 0 and 0 1 The corresponding distribution and random variable are called the Standard Normal Distribution and the Standard Normal Random Variable respectively The Standard Normal Variable will commonly be denoted Z Section 85 The Normal Distribution The area of the region under the standard normal curve to the left of some value 2 Le PZ lt2 or PZ S 2 is calculated for us in Table II Appendex B on pg 1177 Example 1 Let Z be the standard normal variable By rst making a sketch of the appropriate region under the standard normal curve find the values of a PZlt 491 U PZ lt 044 c PZ gt 05 d PZgt 256 D P165 lt zlt 202 139quot P1 lt Zlt 247 Section 85 The Normal Distribution Example 2 Let Z be the standard normal variable Find the value ofz ifz satisfies a PZltz 09495 b PZgt z 09115 c PZlt z 06950 d PZ ltZltZ 07888 e Pz ltZltZ 08444 Section 85 The Normal Distribution When given a normal distribution in which u 9quot 0 and a 9quot 1 we can transform the normal curve to the standard normal curve by doing Whichever of the following applies PXltbPZltb39u PXgtaPzgta PaltXltbPa 39u 17 Example 3 SupposeXis a normal variable with u 80 and a 10 Find a PX lt 100 b PX gt 65 c P70 lt X lt 95 Section 85 The Normal Distribution Example 4 SupposeXis a normal variable with u 7 and a 4 Find a PX lt 11 b PX gt 208 c P 052 lt X lt 384 34 733 32 31 730 29 728 27 726 25 24 723 22 721 20 19 18 l7 v16 15 14 13 712 11 710 09 08 707 06 05 B l TABLES Table 2 The Standard Normal Distribution Fzz PZ s z 000 001 002 003 004 005 006 007 008 009 00003 00003 00003 00003 00003 00003 00003 00003 00003 00002 00005 00005 00005 00004 00004 00004 00004 00004 00004 00003 00007 00007 00006 00006 00006 00006 00006 00005 00005 00005 00010 00009 00009 00009 00008 00008 00008 00008 00007 00007 00013 00013 00013 00012 00012 00011 00011 00011 00010 00010 00019 00018 00017 00017 00016 00016 00015 00015 00014 00014 00026 00025 00024 00023 00023 00022 00021 00021 00020 00019 00035 00034 00033 00032 00031 00030 00029 00028 00027 00026 00047 00045 00044 00043 00041 00040 00039 00038 00037 00036 00062 00060 00059 00057 00055 00054 00052 00051 00049 00048 00082 00080 00078 00075 00073 00071 00069 00068 00066 00064 00107 00104 00102 00099 00096 00094 00091 00089 00087 00084 00139 00136 00132 00129 00125 00122 00119 00116 00113 00110 00179 00174 00170 00166 00162 00158 00154 00150 00146 00143 00228 00222 00217 00212 00207 00202 00197 00192 00188 00183 00287 00281 00274 00268 00262 00256 00250 00244 00239 00233 00359 00352 00344 00336 00329 00322 00314 00307 00301 00294 00446 00436 00427 00418 00409 00401 00392 00384 00375 00367 00548 00537 00526 00516 00505 00495 00485 00475 00465 00455 00668 00655 00643 00630 00618 00606 00594 00582 00571 00559 00808 00793 00778 00764 00749 00735 00722 00708 00694 00681 00968 00951 00934 00918 00901 00885 00869 00853 00838 00823 01151 01131 01112 01093 01075 01056 01038 01020 01003 00985 01357 01335 01314 01292 01271 01251 01230 01210 01190 01170 01587 01562 01539 01515 01492 01469 01446 01423 01401 01379 01841 01814 01788 01762 01736 01711 01685 01660 01635 01611 02119 02090 02061 02033 02005 01977 01949 01922 01894 01867 02420 02389 02358 02327 02296 02266 02236 02206 02177 02148 02743 02709 02676 02643 02611 02578 02546 02514 02483 02451 03085 03050 03015 02981 02946 02912 02877 02843 02810 02776 H77 34 733 32 31 730 29 728 27 726 25 24 723 22 721 20 19 18 l7 v16 15 14 13 712 11 710 09 08 707 06 05 B l TABLES Table 2 The Standard Normal Distribution Fzz PZ s z 000 001 002 003 004 005 006 007 008 009 00003 00003 00003 00003 00003 00003 00003 00003 00003 00002 00005 00005 00005 00004 00004 00004 00004 00004 00004 00003 00007 00007 00006 00006 00006 00006 00006 00005 00005 00005 00010 00009 00009 00009 00008 00008 00008 00008 00007 00007 00013 00013 00013 00012 00012 00011 00011 00011 00010 00010 00019 00018 00017 00017 00016 00016 00015 00015 00014 00014 00026 00025 00024 00023 00023 00022 00021 00021 00020 00019 00035 00034 00033 00032 00031 00030 00029 00028 00027 00026 00047 00045 00044 00043 00041 00040 00039 00038 00037 00036 00062 00060 00059 00057 00055 00054 00052 00051 00049 00048 00082 00080 00078 00075 00073 00071 00069 00068 00066 00064 00107 00104 00102 00099 00096 00094 00091 00089 00087 00084 00139 00136 00132 00129 00125 00122 00119 00116 00113 00110 00179 00174 00170 00166 00162 00158 00154 00150 00146 00143 00228 00222 00217 00212 00207 00202 00197 00192 00188 00183 00287 00281 00274 00268 00262 00256 00250 00244 00239 00233 00359 00352 00344 00336 00329 00322 00314 00307 00301 00294 00446 00436 00427 00418 00409 00401 00392 00384 00375 00367 00548 00537 00526 00516 00505 00495 00485 00475 00465 00455 00668 00655 00643 00630 00618 00606 00594 00582 00571 00559 00808 00793 00778 00764 00749 00735 00722 00708 00694 00681 00968 00951 00934 00918 00901 00885 00869 00853 00838 00823 01151 01131 01112 01093 01075 01056 01038 01020 01003 00985 01357 01335 01314 01292 01271 01251 01230 01210 01190 01170 01587 01562 01539 01515 01492 01469 01446 01423 01401 01379 01841 01814 01788 01762 01736 01711 01685 01660 01635 01611 02119 02090 02061 02033 02005 01977 01949 01922 01894 01867 02420 02389 02358 02327 02296 02266 02236 02206 02177 02148 02743 02709 02676 02643 02611 02578 02546 02514 02483 02451 03085 03050 03015 02981 02946 02912 02877 02843 02810 02776 H77 1178 B I TABLES 04 03 02 01 00 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Table 2 continued Fzz PZ 5 z 000 03446 03821 04207 04602 05000 05000 05398 05793 06179 06554 06915 07257 07580 07881 08159 08413 08643 08849 09032 09192 09332 09452 09554 09641 09713 09772 09821 09861 09893 09918 09938 09953 09965 09974 09981 09987 09990 09993 09995 09997 001 03409 03783 04168 04562 04960 05040 05438 05832 06217 06591 06950 07291 07611 07910 08186 08438 08665 08869 09049 09207 09345 09463 09564 09649 09719 09778 09826 09864 09896 09920 09940 09955 09966 09975 09982 09987 09991 09993 09995 09997 002 03372 03745 04129 04522 04920 05080 05478 05871 06255 06628 06985 07324 07642 07939 08212 08461 08686 08888 09066 09222 09357 09474 09573 09656 09726 09783 09830 09868 09898 09922 09951 09956 09967 09976 09982 09987 09991 09994 09995 09997 003 03336 03707 04090 04483 04880 05120 05517 05910 06293 06664 07019 07357 07673 07967 08238 08485 08708 08907 09082 09236 09370 09484 09582 09664 09732 09788 09834 09871 09901 09925 09943 09957 09968 09977 09983 09988 09991 09994 09996 09997 004 03300 03669 04052 04443 04840 05160 05557 05948 06331 06700 07054 07389 07704 07995 08264 08508 08729 08925 09099 09251 09382 09495 09591 09671 09738 09793 09838 09875 09904 09927 09945 09959 09969 09977 09984 09988 09992 09994 09996 09997 005 03264 03632 04013 04404 04801 05199 05596 05987 06368 06736 07088 07422 07734 08023 08289 08531 08749 08944 091 15 09265 09394 09505 09599 09678 09744 09798 09842 09878 09906 09929 09946 09960 09970 09978 09984 09989 09992 09994 09996 09997 006 03228 03594 03974 04364 04761 05239 05636 06026 06406 06772 07123 07454 07764 08051 08315 08554 08770 08962 09131 09278 09406 09515 09608 09686 09750 09803 09846 09881 09909 09931 09948 09961 09971 09979 09985 09989 09992 09994 09996 09997 007 03192 03557 03936 04325 04721 05279 05675 06064 06443 06808 07157 07486 07794 08078 08340 08577 08790 08980 09147 09292 09418 09525 09616 09693 09756 09808 09850 09884 0991 1 09932 09949 09962 09972 09979 09985 09989 09992 09995 09996 09997 008 03156 03520 03897 04286 04681 05319 05714 06103 06480 06844 07190 07517 07823 08106 08365 08599 08810 08997 09162 09306 09429 09535 09625 09699 09761 09812 09854 09887 09913 09934 09951 09963 09973 09980 09986 09990 09993 09995 09996 09997 009 03121 03483 03859 04247 04641 05359 05753 0614 06517 06879 07224 07549 07852 08133 08389 08621 08830 09015 09177 09319 09441 09545 09633 09706 09767 09817 09857 09890 09916 09936 09952 09964 09974 09981 09986 09990 09993 09995 09997 09998

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