Conditional Expectations
Bivariate Probability Distributions
Discrete Case - Television Sales Data
Data Source: Benson (1966): “Analysis of Irregular Two-Dimensional Distributions of
Consumer Buying Choices”, Journal of Marketing Research, Vol. 3, #3, pp.279-288
Discrete Bivariate Distributions
Joint Probabilit y Distributi on :
p ( x1 , x2 )  P ( X 1  x1 , X 2  x2 )
 p( x , x )  1
1
x1
2
x2
Marginal Probabilit y Distributi ons :
p1 ( x1 )  P ( X 1  x1 )   p ( x1 , x2 )
x2
p2 ( x2 )  P ( X 2  x2 )   p ( x1 , x2 )
x1
 p (x )  1
1
1
x1
 p (x )  1
2
2
x2
Conditiona l Distributi ons (assuming p1 ( x1 ), p2 ( x2 )  0) :
p ( x1 , x2 )
p ( x1 | x2 )  P ( X 1  x1 | X 2  x2 ) 
p 2 ( x2 )
 p( x
p ( x1 , x2 )
p1 ( x1 )
 p( x
p ( x2 | x1 )  P ( X 2  x2 | X 1  x1 ) 
1
| x2 )  1 x2
x1
x2
2
| x1 )  1 x1
Example - Television Sales
Counts (Top Table) - Joint&Marginal Probabilites (Bottom Table)
Frequencies
Size\Price
15.5
17.5
19.5
21.5
23.5
Total
$75
11
20
31
16
3
81
$125
21
123
465
136
36
781
$175
22
166
780
239
104
1311
$225
6
29
168
79
94
376
$275
1
7
34
31
28
101
Probabilities
Size\Price
15.5
17.5
19.5
21.5
23.5
Probability
$75
0.004091
0.007438
0.011528
0.005950
0.001116
0.030123
$125
0.007810
0.045742
0.172927
0.050576
0.013388
0.290443
$175
0.008181
0.061733
0.290071
0.088881
0.038676
0.487542
$225
0.002231
0.010785
0.062477
0.029379
0.034957
0.139829
$275
0.000372
0.002603
0.012644
0.011528
0.010413
0.037560
$325
1
4
8
14
12
39
Total
62
349
1486
515
277
2689
$325
Probability
0.000372
0.023057
0.001488
0.129788
0.002975
0.552622
0.005206
0.191521
0.004463
0.103012
0.014504
1.000000
Example - Television Sales
Conditional Probabilities of Price Given Size
P(Price|Size)
Size\Price
15.5
17.5
19.5
21.5
23.5
$75
0.1774
0.0573
0.0209
0.0311
0.0108
$125
0.3387
0.3524
0.3129
0.2641
0.1300
$175
0.3548
0.4756
0.5249
0.4641
0.3755
$225
0.0968
0.0831
0.1131
0.1534
0.3394
$275
0.0161
0.0201
0.0229
0.0602
0.1011
$325
0.0161
0.0115
0.0054
0.0272
0.0433
Sum
1.0000
1.0000
1.0000
1.0000
1.0000
Conditional Probabilities of Size Given Price
P(Size|Price)
Size\Price
15.5
17.5
19.5
21.5
23.5
Sum
$75
0.1358
0.2469
0.3827
0.1975
0.0370
1.0000
$125
0.0269
0.1575
0.5954
0.1741
0.0461
1.0000
$175
0.0168
0.1266
0.5950
0.1823
0.0793
1.0000
$225
0.0160
0.0771
0.4468
0.2101
0.2500
1.0000
$275
0.0099
0.0693
0.3366
0.3069
0.2772
1.0000
$325
0.0256
0.1026
0.2051
0.3590
0.3077
1.0000
Functions of Discrete Bivariate R.V.s
Function of X 1 , X 2 : g ( X 1 , X 2 )
E g ( X 1 , X 2 )   g ( x1 , x2 ) p ( x1 , x2 )
x1
x2
E g ( X 1 )   g ( x1 ) p ( x1 , x2 )   g ( x1 ) p ( x1 , x2 )   g ( x1 ) p1 ( x1 )
x1
x2
x1
x2
E g ( X 1 ) | X 2  x2    g ( x1 ) p ( x1 | x2 )
x1
x1
Special Cases :
E ( X 1 | X 2  x2 )   x1 p ( x1 | x2 )
x1
V ( X 1 | X 2  x2 )   x1  E ( X 1 | X 2  x2 ) p ( x1 | x2 )
2
x1
  x12 p ( x1 | x2 )  E ( X 1 | X 2  x2 )  E ( X 12 | X 2  x2 )  E ( X 1 | X 2  x2 )
2
x1
Note : Common to write : E X 1 | X 2  x2   E ( X 1 | x2 )
2
Example - Television Sales
Conditional Mean and Variance of Prices for Size = 15.5
Size=15.5
Price(x1) p(x1|15.5) x1*p(x1|15.5) (x1^2)*p(x1|15.5)
75
0.1774
13.3065
997.9839
125
0.3387
42.3387
5292.3387
175
0.3548
62.0968
10866.9355
225
0.0968
21.7742
4899.1935
275
0.0161
4.4355
1219.7581
325
0.0161
5.2419
1703.6290
Sum
1.0000
149.1935
24979.8387
E ( P | S  15.5)   x1 p( x1 | 15.5)  149.1935
x1
V ( P | S  15.5)  E ( P | 15.5)  E ( P | 15.5) 
2
 24979.8387  (149.1935) 2  2721.1328
2
Example - Television Sales
Conditional Mean and Variance of Price for all Sizes
Size(x2)
15.5
17.5
19.5
21.5
23.5
E(P|x2)
149.1935
159.5272
166.0162
176.4563
200.9928
E(P^2|x2)
24979.8387
27329.8711
29104.1386
33702.6699
42989.6209
V(P|x2)
2721.1238
1880.9369
1542.7763
2565.8403
2591.5234
p2(x2)
0.0231
0.1298
0.5526
0.1915
0.1030
Note that the last column is the marginal distribution of sizes
Rules for Conditional Expectations
Rule for Conditiona l/Marginal Means
E ( X 1 )  E X 2 E  X 1 | X 2  x2   E E ( X 1 | X 2 )
Rule for Conditiona l/Marginal Variances
V ( X 1 )  E X 2 V  X 1 | X 2  x2   VX 2 E  X 1 | X 2  x2 
 E V ( X 1 | X 2 )  V E ( X 1 | X 2 )
Example - Television Sales
Marginal Distribution of Prices with Mean and Variance
Price(x1)
75
125
175
225
275
325
Sum
p1(x1)
0.0301
0.2904
0.4875
0.1398
0.0376
0.0145
1.0000
x1*p1(x1)
2.2592
36.3053
85.3198
31.4615
10.3291
4.7136
170.3886
(x1^2)*p1(x1)
169.4403
4538.1647
14930.9688
7078.8397
2840.5076
1531.9357
31089.8568
E ( X 1 )   x1 p1 ( x1 )  170.3886
x1
2


V ( X 1 )   x p1 ( x1 )   x1 p1 ( x1 )  31089.8568  (170.3886) 2  2057.5749
x1
 x1

2
1
Example - Television Sales
Means and Variances of Conditional Distributions and Probabilities
Size(x2)
15.5
17.5
19.5
21.5
23.5
Sum
E(P|x2)
149.1935
159.5272
166.0162
176.4563
200.9928
V(P|x2)
2721.1238
1880.9369
1542.7763
2565.8403
2591.5234
p2(x2)
0.0231
0.1298
0.5526
0.1915
0.1030
1.0000
E(P|x2)p2(x2) (E(P|x2)^2)p2(x2) V(P|x2)p2(x2)
3.4399
513.2169
62.7407
20.7047
3302.9669
244.1231
91.7441
15231.0094
852.5718
33.7951
5963.3571
491.4123
20.7047
4161.4998
266.9587
170.3886
29172.0502
1917.8066
E X 2 E ( P | x2 )  E E ( P | X 2 )   E ( P | x2 ) p2 ( x2 )  170.3886
x2
E X 2 V ( P | x2 )   V ( P | x2 ) p2 ( x2 )  1917.8066
x2


VX 2 E ( P | x2 )   E ( P | x2 )  p2 ( x2 )   E ( P | x2 ) p2 ( x2 )
x2
 x2

 29172.0502  (170.3886) 2  139.7683
2
2
Example - Television Sales
E ( X 1 )   x1 p1 ( x1 )  170.3886
x1


E E ( X 1 | X 2 )    x1 p ( x1 | x2 ) p2 ( x2 )  170.3886
x2  x1

V ( X 1 )   x12 p1 ( x1 )  E ( X 1 )  2057.5749
2
x1
E V ( X 1 | X 2 )   V ( X 1 | x2 ) p2 ( x2 )  1917.8066
x2
2


2
V E ( X 1 | X 2 )   E  X 1 | x2  p2 ( x2 )   E  X 1 | x2 p2 ( x2 )  139.7683
x2
 x2

 E ( X 1 )  E E ( X 1 | X 2 )
V ( X 1 )  E V ( X 1 | X 2 )  V E ( X 1 | X 2 )