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					Calculating expected value, variance, and covariance from a sample (empirical data)
Objectives:
     Practice calculating mean, variance, and covariance.
     Become more familiar with math symbols.
     Better understand the meaning of covariance.

Note that individual answers (observations) are recorded in boxes and class answers (summary statistics) are recorded in circles.

Step 1. Roll your die. Define X = the result of the first die roll (1-6)

     record your observation, Xi here:

Step 2. Roll your die a second time. Define Y = the sum of both rolls (2-12)

     record your observation, Yi here:

Step 3. We will calculate the sample mean of X and the sample mean of Y as a class. (Follow along to make sure you understand how
the calculation is being made). Record these below:

                                      n

                                  X
                                  i 1
                                                  i
     Sample mean of X: X n                           =
                                          n
                                  n

                                 Y
                                 i 1
                                              i
     Sample mean of Y: Yn                        =
                                      n

Step 4: Calculate the squared deviation of your X and Y observations from the group means:

     record your ( X i  X n ) here:
                              2




     record your (Yi  Yn ) here:
                           2



Step 5. We will calculate the sample variance of X and the sample variance of Y as a class. (Follow along to make sure you
understand how the calculation is being made). Record these below:

                                                       n

                                                      (X
                                                      i 1
                                                                 i     X n )2
         Sample variance of X: s x 
                                          2
                                                                                     =
                                                             n 1
                                                       n

                                                       (Y
                                                      i 1
                                                             i        Yn ) 2
         Sample variance of Y: s y 
                                          2
                                                                                =
                                                             n 1


Step 6. Calculate the product of X and Y deviations:

      record your ( Yi  Yn )( X i  X n ) here:

Step 6. We will calculate the sample covariance of X and Y. (Follow along to make sure you understand how the calculation is being
made). Record this below:
                                                                          n

                                                                         (Y
                                                                         i 1
                                                                                 i    Yn )( X i  X n )
         Sample covariance of X and Y: s x , y                                                            =
                                                                                         n 1
For class discussion: What do you note about the relationship between variance and covariance here? How
does this make sense intuitively?

RECORD CLASS OBSERVATIONS HERE.

Observation      Xi                Yi               (X i  X n )2      (Yi  Yn ) 2      ( Yi  Yn )( X i  X n )

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15
Calculating expected value, variance, and covariance of a random variable with a known (a priori) probability
distribution:


Mean (expected value):   E ( x)               x p(x )
                                                all x
                                                         i          i




Variance:  x 2  Var ( x)  E ( x   x ) 2           [x
                                                        all x
                                                                        i     x ] 2 p( xi )  algebra  E ( x 2 )   x  E ( x 2 )  E ( x) 2
                                                                                                                           2




Covariance:
  xy  Cov ( xy )  E[( x   x )( y   y )]          (x
                                                    all x i , y i
                                                                    i         x )( y i   y ) p( xi & y i )  algebra

  E ( xy )   x  y  E ( xy )  E ( x) E ( y )

The covariance of two random variables measures their tendency to vary together, i.e., to co-vary. Where the
variance is the average squared deviation of a random variable from its mean, the covariance is the average of
the products of the deviations of two random values from their respective means.
    In-Class Exercise:
    A Priori version of the dice problem:

    Fill in the following joint probabilities, P(X&Y)

    As before,
    X = the result of the first die roll
    Y = sum of both rolls
                                                    X

                                            1           2             3         4   5   6

                          1        P(X=1&Y=1)=0   P(X=2&Y=1)=0   P(X=3&Y=1)=0


                          2         P(X=1&Y=2)=   P(X=2&Y=2)=0
                                           1/36
                          3

                          4

                          5
Y                         6

                          7

                          8

                          9

                          10

                          11

                          12
   Next, using the above joint probability table,
   Calculate the a priori means of X and Y, variances of X and Y, and covariance of X and Y.

   X          Marginal probability table:
              x        x2         p(x)
              1        1     p(x=1)=1/6
              2        4     p(x=2)=1/6
              3        9     p(x=3)=1/6
              4       16     p(x=4)=1/6
              5       25     p(x=5)=1/6
              6       36     p(x=6)=1/6
                                 1.0
         E(x) =

         E(x2) =

          x 2 =Var(x) = E(x2) – [E(x)]2 =
   Y
           Marginal probability table:
             y        y2          p(y)
             1         1            0
             2         4          1/36
             3         9          2/36
             4        16          3/36
             5        25          4/36
             6        36          5/36
             7        49          6/36
             8        64          5/36
             9        81          4/36
            10       100          3/36
            11       121          2/36
            12       144          1/36

E(y) =

E(y2)=

 y 2 =Var(y) = E(y2) – [E(y)]2 =



E(xy) = (refer to joint probability table)

Cov(xy) = E(xy) – [E(x)][E(y)] =
Answer:

                        X


                1      2      3      4      5      6
          1     0      0      0      0      0      0
          2    1/36    0      0      0      0      0
          3    1/36   1/36    0      0      0      0
          4    1/36   1/36   1/36    0      0      0
          5    1/36   1/36   1/36   1/36    0      0
 Y
          6    1/36   1/36   1/36   1/36   1/36    0
          7    1/36   1/36   1/36   1/36   1/36   1/36
          8     0     1/36   1/36   1/36   1/36   1/36
          9     0      0     1/36   1/36   1/36   1/36
          10    0      0      0     1/36   1/36   1/36
          11    0      0      0      0     1/36   1/36
          12    0      0      0      0      0     1/36
   X

                       x                  x2          p(x)
                       1                   1      p(x=1)=1/6
                       2                   4      p(x=2)=1/6
                       3                   9      p(x=3)=1/6
                       4                  16      p(x=4)=1/6
                       5                  25      p(x=5)=1/6
                       6                  36      p(x=6)=1/6
                                                     1.0

                            x p(x )  (1)( 6 )  2( 6 )  3( 6 )  4( 6 )  5( 6 )  6( 6 ) 
                                                    1       1      1       1   21  1      1
         E(x) =                      i     i                                       3.5
                 all x                                                         6

                            
                                         1      1      1       1       1       1
         E(x2) =        xi p(x i )  (1)( )  4( )  9( )  16( )  25( )  36( )  15.16666
                          2

                  all x                  6      6      6       6       6       6

          x 2 =Var(x) = E(x2) – [E(x)]2 = 15 - (3.5)2 = 2.916666

   Y
                 Marginal probability table:
                   y        y2          p(y)
                   1         1            0
                   2         4          1/36
                   3         9          2/36
                   4        16          3/36
                   5        25          4/36
                   6        36          5/36
                   7        49          6/36
                   8        64          5/36
                   9        81          4/36
                  10       100          3/36
                  11       121          2/36
                  12       144          1/36


           y p(y
                                          1
E(y) =             i        i   )(         )[ 2(1)  3(2)  4(3)  5(4)  6(5)  7(6)  8(5)  9(4)  10(3)  11(2)  12(1)]  7.0
          all y                          36

E(y )=  y i 2 p(y i )  1 [4(1)  9(2)  16(3)  25(4)  36(5)  49(6)  64(5)  81(4)  100(3)  121(2)  144(1)]  54.833
    2

         all y                   36

 y =Var(y) = E(y ) – [E(y)]2 = 54.8333-49 = 5.83333
   2                             2

   Notice: the expected value and variance are exactly twice that of those for X!!
Covariance of X and Y:

                                                     1         2         3          4         5         6
                                           1         0         0         0          0         0         0
                                           2       1/36        0         0          0         0         0
                                           3       1/36      1/36        0          0         0         0
                                           4       1/36      1/36      1/36         0         0         0
                                           5       1/36      1/36      1/36       1/36        0         0
                                           6       1/36      1/36      1/36       1/36      1/36        0
                                           7       1/36      1/36      1/36       1/36      1/36      1/36
                                           8         0       1/36      1/36       1/36      1/36      1/36
                                           9         0         0       1/36       1/36      1/36      1/36
                                          10         0         0         0        1/36      1/36      1/36
                                          11         0         0         0          0       1/36      1/36
                                          12         0         0         0          0         0       1/36



            x y p(x
                                          1
                           i , yi   )      [2  3  4  5  6  7  6  8  10  12  14  16  12  15  18  21  24  27  20 
E(xy) =   all x, y
                     i i
                                         36
          24  28  32  36  40  30  35  40  45  50  55  42  48  54  60  66  72]  987 / 36  27.41666


Cov(xy) = 27.41666 – (3.5) (7.0) = 2.91666


Notice: Covariance of X and Y = Variance of X (they share completely the variance of X).

				
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posted:10/28/2011
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