PRIN OF STATISTICS I
PRIN OF STATISTICS I STAT 211
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This 6 page Study Guide was uploaded by Darien Kutch on Wednesday October 21, 2015. The Study Guide belongs to STAT 211 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 67 views. For similar materials see /class/225762/stat-211-texas-a-m-university in Statistics at Texas A&M University.
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Date Created: 10/21/15
Crib Sheet for Exam 2 Statistics 211 1 Chapter 4 Continuous PDF S PaS X S b bfzdz Fltzgt PltX z E fydy Pa g X g b Fb 7 Fa F X 1 For 0 S p S 1 the 100p7th percentile of a continuous distribution you must solve p Fz for z where z is the 100p7th percentile Ewe m he fltzgtdz EX foo 00 M EKX WQl 02 fom1u2fxd1 Remember EX 7 m2 EX2 7 EX2 a2 11 The Uniform Distribution The family of uniform distributions has the following PDF A S I S B otherwise L fzAB 5H 12 The Exponential Distribution The family of exponential distributions has the following PDF 7 Ae 1207Agt0 I M T 0 otherwise 13 Normal Distribution The Normal or Gaussian distribution has the following PDF fltI M70 2i7rae 22 72 fooltzltoo EX M VX a2 shorthand X N NMa2 X M ZNM0a21 ltIgtZPZS2 2 XZat 039 131 Normal Approximation to the Binomial le 14 N binomialnp and np 2 57a 2 5 then X N N0 np 02 npq PX S I m T N E7 77117 PX 2 I 17 W Pzl S X S 12 w W 7 Gamma Distribution The Gamma function Fa fem za le mdz a gt 0 Some properties 1 2 3 1 gt17 Fa a 71Fa 71 If n is positive integer lquotn n 7 1l re 7 w The Gamma PDF 071671 3 fza 120agt0 gt0 1 71 WWI EXMa VXa2a62 a is the shape parameter and is the scale parameter lf 1 then we call this the Standard Gamma Distribution The Standard Gamma CDF is also known as the incomplete gamma func tions Table A4 gives some common values for this function Let X have a Gamma distribution with parameters a and Then for any I gt 07 the CDF of X is given by PXSIFza Fa when F a is the incomplete gamma function X N gammaa 1 7gt X N exp X N gammaa 2 7gt X N X2v If X is a Chisquare RV7 then 1 is referred to as the degrees of freedom of X Chisquare is important because X N N01 A X2 N X201 The distribution of elapsed time between 2 events in a Poisson process distribution is Exponentiale The Exponential distribution is memoryless 15 The Weibull Distribution The PDF of the Weibull distribution fza zaile wma I 2 0701 gt 076 gt 0 2 mama maewwawe The CDF of the Weibull RV is Fza lieim a 120 The Exponential distribution is a special case of the Weibull distribution X N weibulla 1 A X N exp 16 The Lognormal Distribution A nonnegative RV X is said to have a lognormal distribution if the RV Y lnX has a normal distribution 1 x27r 01 Note that M and a are not the mean and std deviation of X but of ln X The mean and variance are I 70 671nacw2202 I 2 0 EX WW2 VX em e 71 Note that lnX has a normal distribution7 the CDF of X can be expressed in terms of the standard normal CDF Fza PltZ W ltWgt 039 17 The Beta Distribution The Beta distribution is a PDF over a nite interval 1 Fa5 x714 1 Bix 34 f7a757AaBBiAlFaF5ltB Agt 314 14ng 7 7 7 a 702 7 B A2045 EX71L7AB Aa5 VX7 7 a62a61 If A O and B 1 We have the Standard Beta Distribution 2 Chapter 5 Joint Probability Distributions Marginal probability massdensity function pxltxgtZpltxygt pyltygtZpltxygt fxI Mandy fyy mm Y X ltgtltgt 700 Two RV X Y are independent if for every pair of X Y Maw Px pyy ay fXI fyy otherwise they are dependent Conditional probability density function of Y given that X x is f L y fx I 10067 y 1 I fmmz fooltyltoo melI Expected values Emmy ZZhltzygtpltzygt EMILY m m hltzygtfltzygtdzdy 0071093 EKX 7 MMY Myl ElXYl 7 Mm C0vX Y a pm COTTXY 71 pm 1 Central Limit Theorem CLT states that if our random variables X have a dis tribution With Whose mean and variance exist and X1 X2 Xn are a random sample and our sample is large say n 2 30 then 2 7 a X N 1W7 2 Linear combinations Ea1X1 aan a1EX1 anEXn Va1X1aan Z aiajC0vXlXj i1 j1 lf X1 X7 are mutually independent then Va1X1 aan a VX1 ailan since C0vXZXj 0 When ia j and C0vXZXi VXi lf X1 Xn are independent and normal though they may have different M and al for each Xi then any linear combination of Xils also has a normal distribution 3 Chapter 6 Concepts of Point Estimation Method of Moments X17 7 Xn is a random sample from PDF Then the o Klth population moment is EXk o Klth sample moment is eg 17st sample moment E 27nd sample moment i The MOM estimators 177 m are obtained by equating the rst m sam ple moments to the corresponding first m population moments and solving for 17 7 m Maximum Likelihood Estimation ln MLE we choose the parameters from all possible distributions that Will maximize the probability of us being correct The basic steps for finding an MLE estimator 1 Find the Likelihood function Which essentially is the Joint Probability function f11 39f12 f1n fI171271n 9 L I1712 71n 2 Find the maximum of the LogLikelihood by d l L 39 i 0 En 5117127 7 Tn and solve for 9 MLE Properties 0 Invariance Principle lf 317 76m be the MLE s of 917 76m then the MLE of any function ht917 7 9m is the function h 17 7 of the MLE s o MLE s are approximately unbiased for large n o MLE s are close to MVUE of 9 4 Chapter 7 Confidence Intervals Basic con dence interval7 a is known a i i Z042 Large sample Cl 7 s z i Zed2W Cl for p if TL 2 5 and m 2 5 and n is small compared to the population size N 5 i Z042 lln A small sample Cl i i tut27171 3 7 l I itaQmil 8 1 Tl n 71 n 71 J 2 v 2 XuzQmil Mia27171 Prediction interval Cl for 02 Sample size calculations 7 4232154 L2 n lt2Za2gt2 n
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