Editing Conditional expectation (section)

==Examples==

=== Example 1: Dice rolling ===
Consider the roll of a fair die and let ''A'' = 1 if the number is even (i.e., 2, 4, or 6) and ''A'' = 0 otherwise. Furthermore, let ''B'' = 1 if the number is prime (i.e., 2, 3, or 5) and ''B'' = 0 otherwise.
{| class="wikitable"
|-
!  !! 1 !! 2 !! 3 !! 4 !! 5 !! 6
|-
| A || 0 || 1 || 0 || 1 || 0 || 1
|-
| B || 0 || 1 || 1 || 0 || 1 || 0
|}

The unconditional expectation of A is <math>E[A] = (0+1+0+1+0+1)/6 = 1/2</math>, but the expectation of A ''conditional'' on ''B'' = 1 (i.e., conditional on the die roll being 2, 3, or 5) is <math>E[A\mid B=1]=(1+0+0)/3=1/3</math>, and the expectation of A conditional on ''B'' = 0 (i.e., conditional on the die roll being 1, 4, or 6) is <math>E[A\mid B=0]=(0+1+1)/3=2/3</math>. Likewise, the expectation of B conditional on A = 1 is <math>E[B\mid A=1]= (1+0+0)/3=1/3</math>, and the expectation of ''B'' conditional on ''A'' = 0 is <math>E[B\mid A=0]=(0+1+1)/3=2/3</math>.

=== Example 2: Rainfall data ===
Suppose we have daily rainfall data (mm of rain each day) collected by a weather station on every day of the ten-year (3652-day) period from January 1, 1990, to December 31, 1999.  The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts for those 3652 days. The ''conditional'' expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of the ten–year period that fall in March. Similarly, the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall amounts that occurred on the ten days with that specific date.