3.2.2 Weighted Average, Double Expectation, and the Law of Total Variance Flashcards
(17 cards)
What does the ‘weighted average’ method describe in probability?
A method of combining multiple conditional distributions using their associated probabilities as weights.
What is the general formula for the probability function of a mixed distribution?
f(x) = p₁·f₁(x) + p₂·f₂(x) + … + pₙ·fₙ(x), where the pᵢ sum to 1.
What is the key difference between a weighted average of probability functions and a weighted average of random variables?
Weighted average of functions combines distributions; weighted average of variables combines outcomes.
What is the formula for the CDF of a mixed distribution using weights?
F(x) = p₁·F₁(x) + p₂·F₂(x) + … + pₙ·Fₙ(x).
What is the formula for the survival function of a mixed distribution?
S(x) = p₁·S₁(x) + p₂·S₂(x) + … + pₙ·Sₙ(x).
note: not equal to 1-F(x)
What is the formula for the expected value of a mixed distribution (weighted average)?
E[X] = p₁·E[X₁] + p₂·E[X₂] + … + pₙ·E[Xₙ].
What’s the shortcut name for the formula E[X] = E[E[X | Y]]?
The Law of Total Expectation or the Double Expectation Rule.
What’s the formula for the Law of Total Variance?
Var(X) = E[Var(X | Y)] + Var(E[X | Y]).
How do you remember the Law of Total Variance?
Use the acronym EVVE: Expected Value of Variance + Variance of Expected value.
When is it incorrect to average conditional variances directly?
When the conditional expectations differ — this approach underestimates the total variance.
What does it mean if a distribution is ‘mixed’?
It consists of two or more distinct conditional distributions, weighted by some probability rule.
In the coin/die example, why can’t we call the resulting distribution uniform?
Because it mixes two different uniforms (on [1,4] and [1,6]) with unequal weights — the outcome is not uniform over [1,6].
What is the expected value of a mixed variable when outcomes are 0 and an exponential?
E[X] = (weight of 0)·0 + (weight of Exp)·E[Exp] = w₂·θ.
How is the variance of a mixed variable calculated correctly?
Use either: Var(X) = E[X²] - (E[X])² or Var(X) = E[Var(X | Y)] + Var(E[X | Y]).
What is the process for computing E[X²] of a mixed variable?
E[X²] = p₁·E[X₁²] + p₂·E[X₂²] + … .
What are the two sources of variance in the Law of Total Variance?
One from randomness within each conditional distribution, and one from variability between their means.
In insurance risk classification, how does total variance help?
It separates risk from within-group variability and between-group heterogeneity (e.g., high vs. low risk classes).
