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# Mathematical Statistics MT 427

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This 14 page Class Notes was uploaded by Mr. Halie Wilkinson on Saturday October 3, 2015. The Class Notes belongs to MT 427 at Boston College taught by Jenny Baglivo in Fall. Since its upload, it has received 39 views. For similar materials see /class/218064/mt-427-boston-college in Mathematics (M) at Boston College.

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Date Created: 10/03/15

Brief Review of Probability prepared by Professor Jenny Baglivo Copyright 2004 by Jenny Al Baglivoi All Rights Reserved These notes are a brief review of probability theory The complete set of notes for probability theory are at the MT426 website 0 Brief Review of Probability 2 01 Random variable RV PDF CDF quantiles i i i i i i i i i i i i i i i i i i i i i i 2 0 2 Expectation EX VarX SDX i i i i i i i i i i i i i i i i i i i i i i i i i i i i 3 0 3 Bivariate distributions i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 4 04 Linear combinations of independent RVs i i i i i i i i i i i i i i i i i i i i i i i i i 8 05 Linear combinations of independent normal RVs i i i i i i i i i i i i i i i i i i i i 9 06 Central limit theorem i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 11 07 Random samples and sample summaries i i i i i i i i i i i i i i i i i i i i i i i i i 13 0 Brief Review of Probability 01 Random variable RV PDF CDF quantiles A random variable X is a function from the sample space of an experiment to the real numbers The range of X is the set of values the random variable assumes X is said to be 1 a discrete random variable if its range is either a nite set or a countably in nite set 2 a continuous random variable if its range is an interval or a union of intervals or 3 a mixed random variable if it is neither discrete nor continuous Cumulative distribution function CDF The GDP of X is the function PX 3 cc for all real numbers cc If X is a discrete random variable then is a step function if X is continuous then is continuous Probability density function PDF If X is a discrete random variable then the PDF is a probability pc PX cc for all real numbers cc If X is a continuous random variable then the PDF is a rate d fc UTF whenever the derivative exists Quantiles The pth quantile or IOOpth percentile clip is the point satisfying the equation PX S ccp p To nd mp solve the equation p for ac Example 01 Exponential RV Let X be the continuous random variable with PDF fc Ae m when cc 2 O and 0 otherwise where A is a positive constant Given t gt 0 50 Ae Azdw 1 7 a Thus the GDP of X is 1 7 e Az when cc gt O and 0 otherwise Given 0 lt p lt 1 lief p gt at 710g1Pgt where log is the natural logarithm function Thus the pth quantile of X can be written as follows ccp flog1 7p 02 Expectation EX VarX SDX Let gX be a real valued function Then the mean or expectation or expected value of gX can be computed as follows Ex gwpw when X is discrete E9X fz gcfw doc when X is continuous where the sum integral is assumed to be over all cc with nonzero PDF The expectation is de ned as long as the sum integral converges absolutely Properties of sums and integrals imply the following 1 Ea a where a is a constant 2 Ea bX a bEX where a and b are constants 3 EclglX c292X c1Egl c2EggX where c1 and c2 are constants Mean variance standard deviation If X is a random variable with mean a EX then the variance of X is a varltXgt EltltX 7 NZ when this quantity exists In addition the standard deviation of X is the positive square root of the variance 0 SDX VarX Property 1 VarX EX2 7 2 where a Proof Since a is a constant VarXgt EX M2 EX2 2MX M2 EX2gt 2MEX M2 EX2gt 2 Property 2 If Y a bX where a and b are constants then may bzva and SDY bSDX Proof Since a and b are constants varm mm bx e a bigtgt2gt MW 7 NZ b2EltltX 7 NZ Warm The second statement follows using square roots Example 02 Binomial RV Let X be the number of successes in 4 independent trials of a Bernoulli experiment with success probability 020 The PDF of X is pc 020z0804 z when cc O1234 and 0 otherwise The mean and variance of X can be computed as follows EX 0 X 04096 1 X 04096 2 X 01536 3 X 00256 4 X 00016 080 EX2 02 X 04096 12 X 04096 22 X 01536 32 X 00256 42 X 00016 128 VarX EX2 i EX2 128 7 0802 064 Lastly the standard deviation of X is SDX 080 Example 03 Uniform RV Let X be the uniform random variable on the interval 71020 The PDF of X is as follows x when 710 S x S 20 and 0 otherwise The mean and variance of X can be computed as follows EXgt 3710 7 3Tb d7 5 EX2 22710 x2 doc 100 VmX EX2 7 EX2 100 7 52 75 Lastly the standard deviation of X is SDX x 75 5 03 Bivariate distributions A probability distribution describing the joint variability of two or more random variables is called a joint distribution A bivaridte distribution is the joint distribution of a pair of random variables The joint ODF of the random pair X Y is the function Fwy PX S wY S 3 for all real pairs Discrete random pairs The joint PDF of the discrete random pair X Y is de ned as follows pwy PX w and Y y PX wY y for all real pairs We sometimes write prwg to emphasize the two random variables X and Y are said to be independent when prwg pXw pyy for all real pairs wy where pXw and pyy are the marginal frequency functions of X and Y respectively That is PX wY y PX wPY y for all real pairs Continuous random pairs The joint GDP of the continuous random pair XY is de ned as follows a a 7 lt lt 7 lt lt aw awayPXw7Y3 ayawPXw7Y3 when the joint GDP has continuous second partial derivatives The notation nywy is sometimes used to emphasize the two random variables X and Y are said to be independent when nyocy fXw fyy for all real pairs ccy where fXw and fyy are the marginal density functions of X and Y respectively Expectation Let gXY be a real valued function The mean or expected value or expectation of gX Y can be computed as follows Zw y gwypwy in the discrete case E9X7Y 3 gcyfwydcc in the continuous case where the sum integral is over all pairs with nonzero joint PDF The expectation is de ned as long as the sum integral converges absolutely Properties of sums and integrals and the fact that the joint PDF of independent random variables equals the product of the marginal PDFs can be used to prove the following 1 If a In and b2 are constants and g XY are real valued functions 12 then Ea b191XY b292XY a b191XY b292XY 2 If X and Y are independent and gX and hY are real valued functions then E9XhY E9XgtEhYgtgt Covariance and correlation Let X and Y be random variables with nite means az M and nite standard deviations oz oy The covariance of X and Y is de ned as follows CovltXYgt EltltX 7 my 7 it The notation ow 0011X Y is often used to denote the covariance The correlation of X and Y is de ned as follows C X Y COMX7y M am all away 39 The notation o CorrXY is used to denote the correlation The parameter p is called the correlation coe lcient Table 01 Joint and marginal distributions for trinomial example 310 311 312 313 314 w 0 00081 00540 01350 01500 00625 04096 x 00216 01080 01800 01000 04096 x 2 00216 00720 00600 01536 x 00096 00160 00256 x 4 00016 00016 00625 02500 03750 02500 00625 10000 Some properties of covariance are as follows 1 CO UXXgt VarX 2 CO UYXgt CovXY 3 Cava bXc dY deavX Y 4 CavXY EXY 7 5 If X and Y are independent then CavXY 0 Some properties of correlation are as follows 1 71 S CorrXY S 1 2 If a b c and d are constants then CorrXY when bd gt 0 Corrlta bXc dY COTTXYgt when bd lt 0 In particular CorrXa bX equals 1 if b gt 0 and equals 71 if b lt 0 3 If X and Y are independent then CorrXY 0 Example 04 Trinomial Distribution An experiment has three outcomes which occur with probabilities 020 050 and 030 respectively Let X be the number of occurrences of outcome 1 and Y be the number of occurrences of outcome 2 in 4 independent trials of the experiment Then X Y has a trinomial distribution with parameters 71 4 and 221122123 020050030 The joint PDF of XY is as follows pwy w y 4 i w 7 ygt020z050y03047z7y when 563 01234w y S 4 and zero otherwise See Table 01 for joint and marginal probabilities 2 Y 3 5 2 15 l 5 O 39 051152253X Figure 01 Joint PDF for a random pair left and region of nonzero density right Since X is a binomial RV with n 4 and p 020 EX np 080 and VarX 711317 13 064 Since Y is a binomial RV with n 4 and p 050 EY np 2 and VarY 71130713 Further EXY 20 3 w 3 WW 00464 10108 20252 30116 40060 12 These computations imply that 0071XY EXY7EXEY 1207080200 7040 and 0071XY 7040 CorrXY W 7050 Example 05 Tetrahedral Distribution Let XY be a continuous pair whose joint PDF is as follows 2 fwy y 7 w when 0 lt w lt y lt 3 and 0 otherwise The left part of Figure 01 shows the solid of volume 1 under the surface 2 fwy and above the wy plane over the region of non zero density The right part of the gure shows the region of non zero density 0 lt w lt y lt 3 Then 3 y 3 33 dd 7617 Oz0wfw 3 w 3 F027 3 4 7 EX f 2 3 y 2 3 y4 9 EX d d id i 70357000 fw73 w 3 7054 3 10 and VarX EX2 7 34 2780 Similarly 3 3 y 33 g EY dw dy 7 dy 7 y0 390 y 9 4 EltY2gt f7Ox0y2fw7y dw dy if d and VarY EY2 7 94 2780 u Q H U l Further 3 y 3 y4 9 EXY dw dy idylt y0 10 y0 5 These computations imply that outcry EXY7EXEY 7 01125 and 7 CavXY 7 01125 7 1 CWX Y SDXSDY 27802780 a 04 Linear combinations of independent RVs We consider linear combinations of the form Y ab1X1b2X2 ann a and b are constants where 1 The X1 X2 Xn are mutually independent discrete random variables or mutually independent continuous random variables That is Where the X are discrete with joint PDF P5ltw17w27gtw7wn P1w1P2w2 Pnwn for all n tuples of real numbers is the marginal PDF of X2 or where the X are continuous with joint PDF f w175 27w77 n f1ltw1gtf2w2gt fnwngt for all n tuples of real numbers f is the marginal PDF of X2 2 Each Xi has a nite mean and a nite standard deviation Theorem 06 Linear Combinations Under the conditions above Y has nite mean and variance as follows EY a b1EX1 b2EX2 u bnEXn and VarY bgVaMXl bgVaMXg b VaMXn Proof The rst statement follows by the linearity of expectation The second result uses linearity and independence First note that 2 Y 7 Em 7 biog 7 E09 7 i i InaX 7 EltXgtgtltXj 7 My 2 1 271771 Then VMY MO 7 EY2 211 21bibjEXi 7 EX Xj 7 EXjgt 221 21bibjCOWXivXjgt 2 1 bgCovX X by independence 211 bEVaMX Random samples If each Xi has the same distribution the X distribution then X1 X2 Xn is said to be a random sample of size n from the X distribution We also say that X1 X2 Xn are mutually independent and identically distributed or HD random variables Example 07 Sample Sum Let X be a random variable with nite mean and variance let X1 X2 Xn be a random sample of size n from the X distribution and let YX1X2 Xn be the sample sum Then EY nEX and VarY nVarX Example 08 Sample Mean Let X be a random variable with nite mean and variance let X1 X2 Xn be a random sample of size n from the X distribution and let 05 Linear combinations of independent normal RVs The theorem above tells us about summaries of linear combinations of independent random variables but does not tell us about probability distributions If the Xi are normal random variables then the distribution is known Theorem 09 Independent Normal RVs Let X1 X2 Xn be independent normal random variables and let Yab1X1b2X2 annr Then Y is a normal random variable In particular if X1 X2 Xn is a random sample from a normal distribution with mean u and standard deviation 0 then 1 The sample sum is a normal RV with mean 7111 and variance 7102 and 2 The sample mean is a normal RV with mean u and variance oZn Exercise 010 Composite Score The personnel department of a large corporation gives two aptitude tests to job applicants One measures verbal ability the other quantitative ability From many years experience the company has found that the verbal scores tend to be normally distributed with a mean of 50 and a standard deviation of 10 The quantitative scores are normally distributed with a mean of 100 and a standard deviation of 20 and appear to be independent of the verbal scores A composite score C is assigned to each applicant Where C 2 Verbal Score 3 Quantitative Score If company policy prohibits hiring anyone Whose composite score is below 375 What per centage of applicants will be summarily rejected Solution Let V be the verbal score and Q be the quantitative score of an applicant Then the composite score C 2V 3Q has summary measures EC 2EV 3EQ 400 and VarC 22VarV 32VorQ 4000 Further since V and Q are independent normal random variables C is a normal random variable We want PC lt 375 I 375 7 400 x 4000 where is the GDP of the standard normal random variable gt g ltIgt704017 lt1gt040 03446 Thus about 345 of the applicants will be summarily rejected Exercise 011 Sample Mean Let X1 X2 Xn be a random sample from a normal distribution with mean 10 and standard deviation 3 How large must n be in order that PY 3 105 is at least 99 7 Solution The sample mean Y is a normal random variable with summary measures my EX 10 and VarY M 9 71 71 We need to solve 099 PX 3 105 for n M m2099233 gt ne195r44r 9 n x 9 n Since we want the probability to be at least 99 and since 71 must be a whole number our solution is n 196 099 ltIgtlt Exact sums Other situations where we know exact sums are as follows If X1 is a binomial random variable based on m Bernoulli trials with success proba bility 13 X2 is a binomial random variable based on 712 Bernoulli trials with success probability p and X1 and X2 are independent then X1 X2 has a binomial distri bution with parameters 711 712 and p H lf X1 is a Poisson random variable with parameter A1 X2 is a Poisson random variable with parameter A2 and X1 and X2 are independent then X1 X2 has a Poisson distribution with parameter A1 A2 10 lf X1 is a negative binomial random variable with parameters T1 and 13 X2 is a negative binomial random variable with parameters T2 and p and X1 and X2 are independent then X1 X2 has a negative binomial distribution with parameters r1 T2 and p 9quot If X1 has a gamma distribution with parameters 041 and 8 X2 has a gamma distri bution with parameters 042 and 8 and X1 and X2 are independent then X1 X2 has a gamma distribution with parameters 041 042 and 8 r gt 06 Central limit theorem Assume that X1 X2 X3 is an in nite sequence of HD random variables each with the same distribution as X Let SmX1X2 Xm be the sum of the rst m terms and let be the average of the rst m terms If the X distribution has a nite mean and variance then the central limit theorem says that the distributions of the sample sum Sm and the sample mean Xm are approximately normal when m is large The formal statement is as follows Theorem 012 Central Limit Theorem Under the conditions above let Z 7 Sm7mEX 7 Xm7EX m 7 mVarX 7 VarXm be the standardized sum or standardized average Then for any real number x 7rlbgnOOI7Zm S w Qw where is the cumulative distribution function of the standard normal random variable Special cases Special cases of the central limit theorem include 1 Let X be a binomial RV based on 71 trials with success probability 13 If n is large then the distribution of X is approximately normal with u np and o npl 7 p 2 Let X be a negative binomial RV with parameters r and 13 If r is large then the distribution of X is approximately normal with u r1 7pp and o r1 7 pp2 3 Let X be a Poisson RV with parameter A If A is large then the distribution of X is approximately normal with u A and o 4 Let X be a gamma RV with parameters 04 r and 8 lA If r is large then the distribution of X is approximately normal with u 048 rA and o 0482 7 2 Exercise 013 Poisson Process Suppose that alpha particle emissions can be modelled as a Poisson process with 085 emissions per second on average a Let X be the total number of particles emitted in a ten minute period Use the central limit theorem to approximate PX 2 506 b Let X be the time in seconds until the 100th particle is observed Use the central limit theorem to approximate PX S 120 Solution 1 X is a Poisson random variable with A 510 085 X 600 Now PX 2 506 17 PX lt 506 5055 7 510 z 17 Q z 17 Q 7020 Q 020 05793 lt V510 gt gt gt A continuity correction is used since X takes Whole number values only Solution b X is a gamma random variable with 04 100 A 085 and 8 lA 1085 Now PX g 120 N I 120 7100085 N Mme0852 Since X is a continuous random variable no continuity correction is needed gt g lt1gt020 05793 12 07 Random samples and sample summaries Recall that a random sample of size n from the X distribution is a list of n mutually independent random variables each with the same distribution as X Sample mean sample variance lf X1 X2 Xn is a random sample from a distri bution with mean a and standard deviation 0 then the sample mean X is the random variable i l XX1X2 Xn7 the sample variance 32 is the random variable 32 i 1 n X Y 2 7 n 7 1 lt Z gt 21 and the sample standard deviation S is the positive square root of the sample variance The following theorem can be proven using properties of expectation Theorem 014 Sample Summaries If X is the sample mean and 32 is the sample variance of a random sample of size n from a distribution with mean a and standard deviation 0 then 1 a and VarX 7271 2 E32 a2 Note that in statistical applications the observed value of the sample mean is used to estimate an unknown mean a and the observed value of the sample variance is used to estimate an unknown variance 72 Sample correlation A random sample of size n from the joint X Y distribution is a list of n mutually independent random pairs each with the same distribution as X Y lf X1 Y1 X2Y2 XmYn is a random sample of size n from a bivariate distribution with correlation p CorrX Y then the sample correlation R is the random variable 1X2 XX 7 R where X and 7 are the sample means of the X and Y samples respectively Note that in statistical applications the observed value of the sample correlation is used to estimate an unknown correlation 0 Example 015 BrainBody WeightsAllison amp Cicchetti Science 194732 374 1976 libstatcmueduDASL As part of a study on sleep in mammals researchers collected information on the average brain weight and average body weight for 43 different species 13 Figure 02 Log brain weight vertical axis versus log body weight horizontal axis for the brain body study7 with linear prediction equation superimposed Body weight7 brain weight combinations ranged from 005kg014g for the short tail shrew to 25470kg46030g for the Asian elephant The data for man are 62kg1320g 1364lb291b Let X be the common logarithm of body weight in kilograms and Y be the common logarithm of brain weight in grams Sample summaries are as follows 1 Mean log body weight is E 03117387 with a SD of SE 133113 2 Mean log brain weight is 37 1174217 with a SD of 511 109415 3 Correlation between log body weight and log brain weight is r 0951693 Figure 02 compares the common logarithms of average brain weight in grams vertical axis and average body weight in kilograms horizontal axis for the 43 species The linear prediction equation 8 y 37 r 4W i i Sac is superimposed as well as contours of the best tting bivariate normal density

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