24 Conditional Expected Value
Example 24.1 Roll a fair four-sided die twice. Let
1 | 2 | 3 | 4 | ||
2 | 1/16 | 0 | 0 | 0 | 1/16 |
3 | 0 | 2/16 | 0 | 0 | 2/16 |
4 | 0 | 1/16 | 2/16 | 0 | 3/16 |
5 | 0 | 0 | 2/16 | 2/16 | 4/16 |
6 | 0 | 0 | 1/16 | 2/16 | 3/16 |
7 | 0 | 0 | 0 | 2/16 | 2/16 |
8 | 0 | 0 | 0 | 1/16 | 1/16 |
1/16 | 3/16 | 5/16 | 7/16 |
Find
. How could you find a simulation-based approximation?
Find
. How could you find a simulation-based approximation?
Find
for each possible value of .
Find
. How could you find a simulation-based approximation?
- The conditional expected value (a.k.a. conditional expectation a.k.a. conditional mean), of a random variable
given the event , defined on a probability space with measure , is a number denoted representing the probability-weighted average value of , where the weights are determined by the conditional distribution of given . - Remember, when conditioning on
, is treated as a fixed constant. The conditional expected value is a number representing the mean of the conditional distribution of given . - The conditional expected value
is the long run average value of over only those outcomes for which . - To approximate
, simulate many pairs, discard the pairs for which , and average the values for the pairs that remain.
Example 24.2 Recall Example 23.2. Consider two spins of the Uniform(1, 4) spinner and let
Find
.
Find
for each value of .
Find
.
Find
.
Find
for each value of .
24.1 Conditional expected value as a random variable
Example 24.3 Continuing Example 24.1. Let
Let
- The conditional expected value of
given is the random variable, denoted , which takes value on the occurrence of the event . The random variable is a function of . - For a given value
of , is a number. Let denote the function which maps to the number . The random variable is a function of , namely . - Roughly,
can be thought of as the “best guess” of the value of given only the information available from . - Since
is a random variable, it has a distribution. And since is a function of , the distribution of will be depend on the distribution of . However, remember that a transformation generally changes the shape of a distribution, so the distribution of will usually have a different shape than that of .
Example 24.4 Continuing Example 24.2. Consider two spins of the Uniform(1, 4) spinner and let
Find an expression for
.
Find an expression for
.
24.2 Law of total expectation
Example 24.5 Continuing Example 24.3. Let
Find the expected value of the random variable
- Law of total expectation. For any two random variables
and (defined on the same probability space) - Analogous to the law of total probability, the law of total expectation provides a way of computing an expected value by breaking down a problem into various cases, computing the conditional expected value given each case, and then computing the overall expected value as a probability-weighted average of these case-by-case conditional expected values.
Example 24.6 Recall Example 11.4. You and your friend are playing the “lookaway challenge”. The game consists of possibly multiple rounds. In the first round, you point in one of four directions: up, down, left or right. At the exact same time, your friend also looks in one of those four directions. If your friend looks in the same direction you’re pointing, you win! Otherwise, you switch roles and the game continues to the next round — now your friend points in a direction and you try to look away. (So the player who starts as the pointer is the pointer in the odd-numbered rounds, and the player who starts as the looker is the pointer in the even-numbered rounds, until the game ends.) As long as no one wins, you keep switching off who points and who looks. The game ends, and the current “pointer” wins, whenever the “looker” looks in the same direction as the pointer.
We saw in Example 11.4 that the probability that the player who starts as the pointer wins the game is 4/7 = 0.571.
Compute and interpret the expected number of rounds in a game.
Compute and interpret the conditional expected number of rounds in a game given that the player who is the pointer in the first round wins the game. (Do you think this is greater than, less than, or equal to the value from the previous part?)
Compute and interpret the conditional expected number of rounds in a game given that the player who is the looker in the first round wins the game.
Example 24.7 Suppose you construct a “random rectangle” as follows. The base
Find
.
Find
.
Find
for a generic .
Identify the random variable
.
Use LTE to find
.
Sketch a plot of the joint distribution of
.
Sketch a plot of the marginal distribution of
. Be sure to specify the possible values. Is it Uniform?
What would you need to do to find
using the definition of expected value?
24.3 Taking out what is known
Example 24.8 Continuing Example 24.7. Suppose you construct a “random rectangle” as follows. The base
Explain how you could use simulation to approximate
.
Find
.
Find
.
Find
for a generic . How does relate to ?
Identify the random variable
. How does relate to ?
Use LTE to find
.
Find
. Does the sign of the covariance make sense?
- “Taking out what is known (TOWIK)”
- In particular,
, , and . - Intuitively, when we condition on
we treat it as though its value is known, so it behaves like a non-random constant. For example, is the conditional, random variable analog of the unconditional, numerical relationship where is a constant. But note that TOWIK is a relationship between random variables.