Introduction to Mathematical Statistics
Introduction to Mathematical Statistics MATH 309
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This 2 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 309 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/205287/math-309-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.
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Date Created: 09/17/15
STATMATH 309 DISCUSSION 4 TA JingjiangJack Peng Of ce 1275 M807 1300 Universtiy Avenue Email peng statwiscedu Phone 262 1577 Of ce Hour 1130 130 pm Tuesday or by appoitment Websit wwwstatwiscedu peng Discrete Distribution Bernoulli PX107 PX01 6 Binomial ip n coins7 each of which has probability 6 of coming up heads7 and the probability 1 7 6 of coming up tails Let X be the total number of heads Then X is binomialn76 PX k 77301117 Nig 071727 Remark lf X17 X27 7X are independent idential distribution iid of Bernoulli67 Then Y X1 X2 Xn is Binomialn70 777 Geometric X be the number of tails that appears before the rst head PX k 917 N 7k0717 Negative binomial Y be the number of tials that appear before the rth head rk71 PYk T71 6176k7k071727 10 12 777 PossionPY 7k0 Hypergeometric without replacement PX Continous Distribution UniformL7R density function fx R7L7L R Exp fx Ae M7z 2 0PX 2 s 6 One important property of expontial distribution is 77memoryless PY 2 y MY 2 h PY 2 y See Problem 246 Standard normal7 N071 fa eiwzw E R PM 202 Normal NM702 f 7x 6 R 5 w27r i If X N NM7027 then X 7 M039 N071 If X N NM17039127Y NM27Ug7 X and Y are indlependlent7 then X Y N NW1 27012 0 Ammail o Garnrnaoz7 y fx 7 PM e MJ gt 0 3 Examples 1242 Let W N Uniform A compute each offollowing PW 2 57 PW 2 27PW2 S 97PW2 S 2 2243 Let X N Ep4 Compute following PX 2 57 PX gt 75 PX2 gt 9 3244 Establish for which constants c the following are densities a fx ex on 071 and 0 otherwise b fx m on 071 n is a nonnegative integer c f czlZ on 072 d fx csmz on Om2 4246 rnernoryless of exponetial distribution Let Y N Ezp for some A gt 0 Let y7h 2 0 Prove that PY 7 h 2 ylY 2 h PY 2
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