Statistics for Research I
Statistics for Research I STAT 537
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This 28 page Class Notes was uploaded by Kamren McLaughlin on Monday October 26, 2015. The Class Notes belongs to STAT 537 at University of Tennessee - Knoxville taught by James Schmidhammer in Fall. Since its upload, it has received 26 views. For similar materials see /class/229891/stat-537-university-of-tennessee-knoxville in Statistics at University of Tennessee - Knoxville.
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Date Created: 10/26/15
Discrete Random Variables Def 3 of a discrete random variable Consider an experiment having a nite or countably in nite number of possible outcomes Assign some real number to each possible outcome and associate this number with some variable say X Then X is called a random variable Note that each time the experiment is conducted X may assume a possibly different value re ecting the different outcomes of the experiments Example Flipping a coin Two possible outcomes Heads and Tails Assign the number 1 to heads 0 to tails and associate this with X Thus X 1 if heads O iftails Example Tossing a die Six possible outcomes l 2 3 4 5 6 Directly associate these numbers with X Thus X count on top face of die Example Using the telephone Imagine dialing a certain number using a telephone Let X number of attempts necessary to get a ring at the other end The alternative would be to get a busy signal Possible outcomes 1 2 3 Def 3 of the probabilitv function of a discrete rv Let X be a discrete random variable assuming possible values x1x2 Let f x PX x denote the probability that XX Then f x is called the probability function of the discrete random variable X Properties of f x 1 f x20 2 Zfx1 all x Example Flipping a coin If the coin is fair then PX1PXO forx0l otherwise Example Tossing a die If the die is fair then forx l6 otherwise Question 1 If we toss a die what value would you expect to see on the top face Question 2 If we toss a die what value would we expect to see on the average on the top face Def 3 Mean of a discrete random variable X measure of central tendency u EX foltxgt 2 Def 3 Variance of a discrete random variable X measure of variability 52 VarX EX 2 ice MW 2 x2fx xfx2 EX2 EX2 Def 2 Standard Deviation of a random variable X 62 mew measure of lack of 3 3 2x u fx symmetry x Def 2 Kurtosis of a discrete random variable X measure of heavy H4 Z x 4fx tailedness Def 3 Cumulative Distribution Function of a discrete rand variable X Fx PX g x Zfy ny Some Discrete Distributions Bernoulli Trials A Bernoulli Trial is an event that has two possible outcomes success or failure yes or no Denote the outcome of the Bernoulli Trial by the random variable X with X O for failure 1 for success Denote the probability of a success on a trial by p nggl Then the probability of a failure on a trial is q1 p EX0q1pp EXfox VarX02qr12p I92 p p2p1 ppq VarX EX2 EX gt12 The Binomial Distribution Let the random variable X denote the number of successes in n independent Bernoulli trials with constant probability of success 9 from trial to trial Then X has a binomial distribution with parameters 11 and 9 Its probability function is derived as follows We want fx PX x Thus in 11 trials we have x successes and n x failures One such arrangement is 55S I I l I x n x The probability of observing any particular arrangement of x successes and n x failures is px1 p n n Also there are ways of arrang1ng x x xn x successes and n x failures in a sequence of 11 trials So fltxgt pxlt1 pgt forxo12n x 0 otherwise EltXgt ix x x0 xn x px1 p quot x 1n xp q n71 y niliy np pq 7yx 1 gown 1w np VarX quotpa p VarX EX2 EX2 22x2 F0 xn x x nix pq nP2 n I x2x n x nix E xn xpq n x nix 2 x p 61 quot192 x0 xn x 2 77 2 x72 nix nn 1p q 1119 quot192 H 2 11 11 1 2 n y niziy p gyKn Z y p q 1119 quot192 nn 1I92 quotI quot192 11sz np2 np n1v2 quot190 p Parameter Estimation Point Estimation Let X1Xn be a random sample from some population characterized by the unknown parameter 6 Let 6 hX1Xn be some f ofthe sample data is a statistic which is an estimator of 6 Example X1 iidNn1 i1 n 7 l X ZXZ estlmates LL 11 Does s2 LIZQQ if estimate LL n Properties of estimators l Unbiasedness EG 6 Example X1 iid 62i ln n NI 1 Xi il 150 13le 7 11 1n EX 1 1n ZZ1H 1 H u 22 quot X X2 S n IZZIXZ 1 n 2 E 2 E X X s n1 1 n E X2 2XX X2 11 1 1 1 E n Xf nf quot 1 i1 1 62Ex MZEx2 p2 2 EX2 EX2 39 I quot 1 I n 6 EfH2Ef2 u2 1 2 2 2 52 26 2 6 7l LL 7 n 2 G 2 Ef ciency Let 61 and 62 be estimators of 6 Let MSE E m2 E E 2 e E 2 A A 2 Vare Biase Then 61 is more ef cient than 62 if MSE lltMSE z Relative ef ciency of 62 to 61 MSE 62 Example X1 iid u i1n MSElt l 111 2X1 112 Z E 1u EfLL MSE 1 Varx1 62 MSE Var G72 Method of Maximum Likelihood Let X1Xn be a random sample from some d f x with 6 an unknown parameter The likelihood f 3 of the sample is MG HfOCZ il The maximum likelihood estimator MLE of 9 is that value of 6 that maximizes L6 W Let XlXn be iid from fx91779k De ne sample moments about the origin so 1 z mlz le I1k 7 11 Set u m in the end of k eq E in k unknowns Hfelaluaek 7 Solve for 616k Interval Estimation We wish to construct an interval estimate of the unknown parameter 6 of the form L g 6 s U such that PL 6 Ul oc for some chosen on This is called a 1001 oc con dence interval Twosided con dence interval L g 6 s U Onesided con dence interval L g 6 lower CI 6 s U upper CI CI on LL62 known Let X1Xn iidu62 Then Z XHNO1 so 6 n Ti 2042 3 Z s Zoe2 1 OL where Zoe2 is the quantile from NO1 such that if Z NO1 then Pz 2 2062 2 3 2 XV G G P gX g Wu 20W So a lOOl oc CI on u 62 known is G XiZOCZ J2 Of 6 G X ZocZ a XZoc2E Interpretation If one repeatedly drew samples of size n from the population and with each sample constructed a lOOloc con dence interval for u then lOOloc of these con dence intervals would contain the true parameter u Another interpretation I estimates LL but with some error Ezlf ul Using 1001 oc CI we are sure that this error is o 5 If we want to estimate LL by I and be 1001 00 con dent that the error is less than or equal to some speci ed error E0 choose less than Zoe2 with con dence 1001 0L CI on LL62 unknown Let X17 Xn N iide gz Xu Recall Z NN 071 SN lt gt n 2 Also in 11 5 n 2 2 Now izzgruy E Us 39l l 2 h 22 X X w ere s Since I and s2 are independent 11 ls2 2 2 N X7271 So A 1001 oc CI for Lt is S Xill ocni 2 where t Distribution with 2 df r4 73 72 r1 0 1 2 4 t Dn with 2 df compared with ND1 r4 73 72 r1 0 1 2 4 t Dn with 2 df Standard Normal Dn t Distribution with 10 df CI on 62 H unknown X1Xn iidNu62 n l S2 NX311 A 1001 oc CI on 62 is given by n 1S2 n ls2 2 7 2 Exnil ligxnil 2 2 ChiSquare Distribution with 5 df ul20025 0 08312 5 10 1283 15 20 25 30 CI on the population proportion 75 We assume that 1 it Nn n 75 true for large n 11 So that 71 75 P ows 2042 1 oc 751 75 n We need to solve fr n Z 751 75 042 n for 7 which yields the following quadratic equation in 7 A 2 2 l Tc Tc izaZ Q 2 2 Z Z H jnz 2f9 2cfc2 O n n Variables and Their Frequency Distributions Points for discussion Variation Populations vs Samples Frequencies and their tabulation into a frequency distribution Sampling Random Representative Variables A characteristic that can be quanti ed in some way Types of Variables Categorical Counting Measurement Nominal Ordinal Interval amp Ratio Frequency Distributions for Categorical Variables Absolute Frequencies Relative Frequencies Bar Graphs Frequency Distributions for Measurement Variables Stern and Leaf Plots Histograms Population Frequency Curves and their Shapes Normal Symmetric Skewed Bimodal Shorttailed or Longtailed
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