Class Notes (806,882)
Statistics (474)
STAT 231 (84)
Lecture

# Week1-Lecture2.pdf

11 Pages
117 Views

School
University of Waterloo
Department
Statistics
Course
STAT 231
Professor
Matthias Schonlau
Semester
Fall

Description
Introduction  to  Probability  models       • Paul  the  octopus.  Does  he  have  ESP?     • Do  the  students  of  STAT  231  have  ESP?       Example:     A  coin  is  tossed  10  times,  and  a  student  is   asked  to  guess  the  outcome  of  the  coin  toss.       Problem:  The  response  variate  is  y,  the   correct  number  of  guesses,  and  the  attribute   of  interest  is  the  average  #  of  correct  guesses   over  all  STAT  231  students  and  all  repetitions       Let  us  assume,  for  the  time  being,  that  there  is   no  ESP.    How  can  we  describe  the  outcomes?       Y  =  random  variable  with  11  possible  values     We  can  construct  the  probability  distribution   of  Y     Y   P(Y)   0     1   2   3   4   5   6   7   8   9   10     E(Y)=?     V(Y)=?     Sd(Y)=?             Problem:  Is  the  average  number  of  correct   guesses  more  than  5?     Plan:  Select  25  students  from  the  class,  and   have  each  person  carry  on  the  experiment.   Record  the  variate  y ifor  each  i=1,2,…25     Data:  y ,  y ,  y ,…y ,  and  compute  the  average     1 2 3 25   Analysis:     Suppose  that    .5   Is  that  large?     € Probability  model:     Suppose  that  the  observation  yi  is  an   independent  realization  of  the  random   variable  Y  or  equivalently,  the  data  y , 1 y ,2   y 3…y 25s  a  realization  of  Y 1  2 , 3 Y , 25Y   Where  each  Y ihas  the  same  probability   function  and  Y , 1 Y ,2Y  25e  independent.     Let    =  1Y  2Y  +…Y25/25,  so  that,  if  the   model  applies,  the  observed  average   y = 5.5  is   a  realization  of   .   €   To  answer  the  question,  we  calcul€te     € P(  ≥  5.5)     Conclusion:   €   What  should  we  conclude  from  the  above   result?     What  if  the  probability  is  “too  high”?  What  is   the  probability  is  “too  low”?                       Summary:     • To  make  a  meaningful  analysis,  we  have   to  know  the  distribution  of  Y  and  the   distribution  of   Y ,  so  that  we  can  compute   the  probabilities  (STAT  230)     • Careful  about  the  difference  between   random  variable  and  its  realization     • How  low  is  too  low  to  conclude  that  the   students  have  ESP?                             Review  of  STAT  230     • A  Random  variable  is  a  function  from   the  sample  space  (Ω)  to  the  real   numbers     • For  a  random  variable  Y  with  support   S  (the  possible  realizations  of  Y),    we   specify     For  Y  discrete,  the  probability   function         f(y)  =  P(Y=y)    y  ε  S     For  Y  continuous,  the  probability     density  function     f(y),  y  ε  S,  where       P(a≤  Y≤  b)  =   ∫ bf (y)   a     • Expected  value  and  standard  deviation:     yf (y) µ  =  E[Y]  = y∈S             or  ∫ yf (y)   € y∈S   sd(Y)  =   Var[Y]   €     • If  Y1, 2Y ,…..Y n  are  independent  and  Y  his   probability  (density)  function  f (y).  Thiir i   joint  probability  (density)  function  is     f (y )*f (y2)*……*f (y )   1 1 2
More Less

Related notes for STAT 231

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.