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Lecture 6

by: Debra Tee

Lecture 6 STATS 250

Debra Tee
GPA 3.85

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Random Variables, Discrete vs Continuous, Distribution of Random Variable. Understanding rules for probabilities of discrete variables, Expectations of random variables, Binomial random variables.
Introduction to Statistics
Brenda Gunderson
Class Notes
random, variables, distribution
25 ?




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This 4 page Class Notes was uploaded by Debra Tee on Monday October 3, 2016. The Class Notes belongs to STATS 250 at University of Michigan taught by Brenda Gunderson in Fall 2016. Since its upload, it has received 4 views. For similar materials see Introduction to Statistics in Statistics at University of Michigan.


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Date Created: 10/03/16
Lecture  6:  Random  Variables     8.1  Random  Variables     -­   Definition:   A  random  variable  assigns  a  number  to  each  outcome  of  a  random   circumstance,  or,  equivalently,  a  random  variable  assigns  a  number  to   each  unit  in  a  population.     -­   Two  broad  classes  of  random  variables:  discrete  random  variables   and  continuous  random  variables.   -­   Definitions:  A  discrete  random  variable  can  take  one  of  a  countable   list  of  distinct  values.  Example:  Number  of  years  of  studies.   -­   A  continuous  random  variable  can  take  any  value  in  an  interval  or   collection  of  intervals.  Example:  Weight  of  a  group  of  people     Distribution  of  Random  Variable     -­   it  is  a  model  that  shows  us  what  values  are  possible  for  that  particular   random  variable  and  how  often  those  values  are  expected  to  occur   (i.e.  their  probabilities).  The  model  can  be  expressed  as  a  function  or   table  or  picture,  depending  on  the  type  of  variable  it  is.         8.2  General  Discrete  Random  Variables   -­  A  discrete  random  variable,  X  ,  is  a  random  variable  with  a  fini te  or   countable  number  of  possible  outcomes.     -­  X  =  the  random  variable.   -­  k  =  a  number  that  the  discrete  random  variable  could  assume.   -­  P(X  =  k)  is  the  probability  that  the  random  variable  X  equals  k.   -­  The  probability  distribution  function  (pdf)  for  a  discrete  random  variable  X   is  a  table  or  rule  that  assigns  probabilities  to  the  possible  values  of  the  X.   -­  One  way  to  show  the  distribution  is  through  a  table  that  lists  the  possible   values  and  their  corresponding  probabilities     Two  conditions  that  must  always  apply  to  the  probabilities  for  a   discrete  random  variable  are:   Condition  1:  The  sum  of  all  of  the  individual  probabilities  must  equal  1.   Condition  2:  The  individual  probabilities  must  be  between  0  and  1.   ▯ -­  A  probability  histogram  or  better  yet,  a  probability  stick  graph,  can  be   used  to  display  the  distribution  for  a  discrete  random  variable.     -­   The  x-­‐axis  represents  the  values  or  outcomes.   -­   The  y-­‐axis  represents  the  probabilities  of  the  values  or  outcomes.   -­   The  cumulative  distribution  function  (cdf)  for  a  discrete  random   variable  X  is  a  table  or  rule  that  provides  the  probabilities  P(X  =  k)  for   any  real  number  k.     -­   Generally,  the  term  cumulative  probability  refers  to  the  probability  that   X  is  less  than  or  equal  to  a  particular  value.         8.3  Expectations  for  Random  Variables     Definition:   -­   The  expected  value  of  a  random  variable  is  the  mean  value  of  the   variable  X  in  the  sample  space,  or  population,  of  possible  outcomes.   Expected  value,  denoted  by  E(X),  can  also  be  interpreted  as  the   mean  value  that  would  be  obtained  from  an  infinite  number  of   observations  on  the  random  variable.   -­   ???? = ???? ???? =   ???? ▯ ▯  -­   ????????????????????????????????  ????????  ????  ????????  ???? ???? =  ???? =   (???????? − ????) ????▯   -­   ????????????????????????????????  ???????????????????????????????????? = ???????????????? ????????????????????????????????     8.4  Binomial  Random  Variables     A  binomial  experiment  is  defined  by  the  following  conditions:   1.  There  are  n  “trials”  where  n  is  determined  in  advance  and  is  not  a   random  value.   2.  There  are  two  possible  outcomes  on  each  trial,  called  “success”  (S)  and   “failure”  (F).   3.  The  outcomes  are  independent  from  one  trial  to  the  next.   4.  The  probability  of  a  “success”  remains  the  same  from  one  trial  to  the   next,  and  this  probability  is  denoted  by  p.  The  probability  of  a  “failure”  is  1   –   p  for  every  trial.     A  binomial  random  variable  is  defined  as  X  =  number  of  successes  in  the  n   trials  of  a  binomial  experiment.   Binomial  Distribution:   -­  P(X=k)  =  ▯ ???? ▯ 1 − ???? ▯▯▯   ▯ ▯ ▯! where   ▯  ????????  ????  ????ℎ????????????????  ???? =   ▯! ▯▯▯ ! -­  If  X  has  binomial  distribution  Bin(n,p),  then   -­  Mean  of  X  is  E(X)  =  np   -­  Standard  Deviation  of  X  is  σ  =  sqrt(np(1-­p))    


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