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Pۘ P (    0pu  SJoint Probability Mass Function 2 (  0u06 |RLet X and Y be two discrete rv s defined on the sample space of an experiment. The joint probability mass function p(x, y) is defined for each pair of numbers (x, y) by 2    I        '     `  c $A ??pP    0$ @  ;Let A be the set consisting of pairs of (x, y) values, then2< 2  7 `   c $A ?? P H  0޽h ? 3380___PPT10. f `f(    0 0: d$Marginal Probability Mass Functions $% 2#($  0Pp XThe marginal probability mass functions of X and Y, denoted pX(x) and pY(y) are given byY 2$#$$$$$ $$ $$$$$ $$$$,<`  c $A ??   H  0޽h ? 3380___PPT10.  %p (    0t  pp V"Joint Probability Density Function# 2#(  0%0 LLet X and Y be continuous rv s. Then f (x, y) is a joint probability density function for X and Y if for any two-dimensional set A  2           "        g`  c $A ??P p    0H5 ]  s%If A is the two-dimensional rectangle2& 2  " `  c $A  ?? W  `   c $A  ?? @   H  0޽h ? 3380___PPT10. I    x (  x` x c $A 1?? g q  1 x 0?   t&= Volume under density surface above A2' 2%   RB x s *D>RB x@ s *D> LB x c $D rr  x <V"`@@RB  x s *DV@ RB x s *DV @RB x s *DV@@RB x s *DV XB x 0DV RB x s *DV@@RB x s *DV@ RB x s *DV @XB x 0D8c 0 x  BC@DE(FAA8c @L0 ph  @   "` @ x # BC@DE(F8c @L0 ph  @   "`@ XB x@ 0DV P ` x c $A 2??@P P 2 x 0I`pd dA = shaded rectangle4 2H x 0޽h ? 3380___PPT10. pUo o(    0S   Z&Marginal Probability Density Functions' 2'(   0W0 ]The marginal probability density functions of X and Y, denoted fX(x) and fY(y), are given by ^ 2 &                   ,?`  c $A  ??pq  H  0޽h ? 3380___PPT10. nJ H@ (    0l  PIndependent Random Variables 2(u  0p`3 [Two random variables X and Y are said to be independent if for every pair of x and y values\ 2           `  c $A  ??     0|{=  when X and Y are discrete orL 2     `  c $A ??P      0聼0 p Omwhen X and Y are continuous. If the conditions are not satisfied for all (x, y) then X and Y are dependent.n 2    @          H  0޽h ? 3380___PPT10. |\ 80 (     0Pp  RMore Than Two Random Variables 2(  0xP` If X1, X2,& ,Xn are all discrete random variables, the joint pmf of the variables is the functionc 2            R (, -"`   c $A ??{$   0  $If the variables are continuous, the joint pdf is the function f such that for any n intervals [a1,b1], & ,[an,bn],s 2?                     ,+@`   c $A ?? p  `   c $A ??  H  0޽h ? 3380___PPT10. OjR $R(  $ $ 0 Z ZIndependence  More Than Two Random Variables. 2.( $ 0P The random variables X1, X2,& ,Xn are independent if for every subsetF 2               ($ $ 0ȼP  pof the variables, the joint pmf or pdf of the subset is equal to the product of the marginal pmf s or pdf s.$m 2k (P7` $ c $A ??`@  H $ 0޽h ? 3380___PPT10. Zr (r(  ( ( 0Ӽ T Conditional Probability Function! 2!( ( 0ؼ` `  Let X and Y be two continuous rv s with joint pdf f (x, y) and marginal X pdf fX(x). Then for any X value x for which fX(x) > 0, the conditional probability density function of Y given that X = x is 2    '                         B  b 'O` ( c $A  ??p     ( 0  If X and Y are discrete, replacing pdf s by pmf s gives the conditional probability mass function of Y when X = x.xs 2    2 5   ,#BH ( 0޽h ? 3380___PPT10. z, ,(    68c"`  05.2 Expected Values, Covariance, and Correlation41(20,<<H  0޽h ? 3380___PPT10. Pۘ jb ,(  , , 04   BExpected Value 2( , <  y <   , 0`  PLet X and Y be jointly distributed rv s with pmf p(x, y) or pdf f (x, y) according to whether the variables are discrete or continuous. Then the expected value of a function h(X, Y), denoted E[h(X, Y)] orT 2    &    Q                 >#  , 0' ]  6is 2 ` , c $A 9?? 6 9  , 08+P   <discrete 2    , 0/ p > continuous 2  `  , c $A 4??@P  4H , 0޽h ? 3380___PPT10. A tl 0(  0 0 07: > Covariance 2 ( 0 <<zI <  \ 0 0X? p- TThe covariance between two rv s X and Y isj+ 2        ` 0 c $A <??p  < 0 0Fp  <discrete 2    0 0J m  > continuous 2  `  0 c $A :??P :H 0 0޽h ? 3380___PPT10. o"P 2*4(  4 4 0dSp` T Short-cut Formula for Covariance! 2!(` 4 c $A !??`0F !f 4 0V"`` H 4 0޽h ? 3380___PPT10. 0d @8 8(  8 8 0\ ? Correlation 2 ({ 8 0`p ?The correlation coefficient of X and Y, denoted by Corr(X, Y), @ 2           3  8 0jp  > defined by 2  ` 8 c $A "??`  "` 8 c $A #?? `@  #H 8 0޽h ? 3380___PPT10. ]m 0<m(  < < 0s@0j KCorrelation Proposition 2( < 0w`D<4___PPT9 n If a and c are either both positive or both negative, Corr(aX + b, cY + d) = Corr(X, Y) For any two rv s X and Y,  2    :           bA 8X` < c $A W??0   WH < 0޽h ? 3380___PPT10. kM @@(  @ @ 0А KCorrelation Proposition 2( @ 0p D<4___PPT9 r If X and Y are independent, then but does not imply independence. for some numbers a and b with  2         8X` @ c $A [??g [` @ c $A Y?? 7 Y` @ c $A ^??`  ^` @ c $A a??v  aH @ 0޽h ? 3380___PPT10. kM< P<(    6ث8c"`  @5.3 Statistics and their Distributions4A(20<<<H  0޽h ? 3380___PPT10. Pۘ PH`D(  D D 0еpP : = Statistic 2 ( D 00PP  QA statistic is any quantity whose value can be calculated from sample data. Prior to obtaining data, there is uncertainty as to what value of any particular statistic will result. A statistic is a random variable denoted by an uppercase letter; a lowercase letter is used to represent the calculated or observed value of the statistic.6R 2  G H D 0޽h ? 3380___PPT10. چ| pH|(  H H 0P  BRandom Samples 2( H 0yp PH@___PPT9" 4The rv s X1,& ,Xn are said to form a (simple random sample of size n if The Xi s are independent rv s. Every Xi has the same probability distribution.$H 2S 2                   !    '  P>48XH H 0޽h ? 3380___PPT10. (f Lf(  L L 0   JSimulation Experiments 2( L 0PG d0The following characteristics must be specified:1 21 L 6V"``p D<4___PPT9 p The statistic of interest. The population distribution. The sample size n. The number of replications k.Nq 2N     8XH L 0޽h ? 3380___PPT10.  > >(    68c"``W  B5.4 The Distribution of the Sample Mean4C(20><<H  0޽h ? 3380___PPT10. Pۘ ME P(  P P 0  0 IUsing the Sample Mean 2( P 0 c zLet X1,& , Xn be a random sample from a distribution with mean value and standard deviation Thenjj 2        ^  ^` P c $A! b??P Q& b` P c $A e??`i e` P c $A" f??0  fr  P 0   BIn addition, with To = X1 +& + Xn," 2             `  P c $A$ g??  gH P 0޽h ? 3380___PPT10. \q< JB T(  T T 0$pp* RNormal Population Distribution 2( T 08)  PTLet X1,& , Xn be a random sample from a normal distribution with mean value and standard deviation Then for any n, is normally distributed, as is To. 2        m  -        ` T c $A! h??PPZp h` T c $A i??0 P  i`  T c $A% j??@  jH T 0޽h ? 3380___PPT10. YV  !   X(  X X 0; MThe Central Limit Theorem 2( X 0?P0  $TLet X1,& , Xn be a random sample from a distribution with mean value and variance Then if n sufficiently large, has approximately a normal distribution with 2        W  E    ` X c $A% k??  p k` X c $A! l??0 *P l` X c $A# o??P; o`  X c $A& r??0 P S  r  X 0 30, the Central Limit Theorem can be used.22 2  . H \ 0޽h ? 33___PPT10i. Zx+D=' = @B + "`(  ` ` 00j V"Approximate Lognormal Distribution# 2#( ` 0P0  \Let X1,& , Xn be a random sample from a distribution for which only positive values are possible [P(Xi > 0) = 1]. Then if n is sufficiently large, the product Y = X1X2& Xn has approximately a lognormal distribution.N 2        U      $             ,   > WC-H ` 0޽h ? 3380___PPT10. %ҰG G(    6xu8c"``W  K5.5 The Distribution of a Linear Combination4L(20G<<H  0޽h ? 3380___PPT10. Pۘ ~vd(  d d 0P@  FLinear Combination 2(d d 0p0S Given a collection of n random variables X1,& , Xn and n numerical constants a1,& ,an, the rv\ 2                      ,/(T d 0hp p  Vis called a linear combination of the Xi s.`, 2        &` d c $A (??@>  (H d 0޽h ? 3380___PPT10. @h 6. h(  h h 0 0Z Z&Expected Value of a Linear Combination' 2'( h 0@P0P Let X1,& , Xn have mean values and variances of respectivelyxp 2        c    d` h c $A+ {??pP {` h c $A* ~??pv ~0 h 0D@P  PWhether or not the Xi s are independent,B) 2     `  h c $A- ??   `  h c $A, ??   H h 0޽h ? 3380___PPT10. M   l(  l l 0l: T Variance of a Linear Combination! 2!(` l c $A. ??; N l 0P0 8If X1,& , Xn are independent,x 2            ` l c $A/ ??@ {   l 0 Pp]  7and 2 `  l c $A0 ?? P H l 0޽h ? 3380___PPT10. W 0(    0: T Variance of a Linear Combination! 2!(`  c $A1 ??`I  8  0$`  "For any X1,& , Xn,x 2           H  0޽h ? 3380___PPT10. W A9@p(  p p 0, @ ['Difference Between Two Random Variables( 2((` p c $A $??` $ p 0@`  "and, if X1 and X2 are independent,h# 2         ` p c $A '??P @  'H p 0޽h ? 3380___PPT10. Ю,  ,$Pt(  t t 0 ^*Difference Between Normal Random Variables+ 2+( t 0!  ^If X1, X2,& Xn are independent, normally distributed rv s, then any linear combination of the Xi s also has a normal distribution. The difference X1  X2 between two independent, normally distributed variables is itself normally distributed. 2             O    3        Y > '%H t 0޽h ? 3380___PPT10. 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