Tag Archives: Indicator Variable

Conditional Variance

If X is a random variable that depends on another random variable I, then

Var(X) = E_I[Var_X(X|I)] + Var_I(E_X[X|I])

This is called the double expectation formula.  It is important to keep track of which random variable in a problem is X and which one is I.  Wieshaus calls I the indicator variable.  In the above equation, Var_X(X|I) and E_X[X|I] are functions of I

Example 1:  Noemi and Harry work at Starbucks.  Noemi’s tip jar contains 30% dollars, 30% quarters, 20% dimes, 10% nickels and 10% pennies.  Harry’s tip jar contains 5% dollars, 10% quarters, 10% dimes, 35% nickels and 40% pennies.  A customer steals a coin from Harry’s jar with 99% probability and from Noemi’s jar with 1% probability.  What is the variance of the stolen amount?

  1. Identify the random variables.
    • The stolen amount is what we’re interested in so this is X.
    • The distribution of X depends on which jar the coin came from so the choice of jar is the indicator variable I.
  2. Find the distribution of E_X
    • E_X[X|I=H] = 0.1065 with 99% probability.
    • E_X[X|I=N] = 0.4010 with 1% probability.
  3. Var_I(E_X[X|I]) = 0.000858629
  4. Find the distribution of Var_X(X|I)
    • Var_X(X|I=H) = 0.04682275 with 99% probability.
    • Var_X(X|I=N) = 0.16020900 with 1% probability.
  5. E_I[Var_X(X|I)] = 0.04795661
  6. Var(X) = 0.000858629 + 0.04795661 = 0.0488152

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