Confusing independent and mutually exclusive events

Independent events satisfy P(A and B) = P(A)*P(B), while mutually exclusive events satisfy P(A and B) = 0. These are very different conditions. Two events (with positive probability) cannot be both independent and mutually exclusive.

Forgetting to check support when computing densities

When finding the PDF of a transformed variable Y = g(X), you must also transform the support. If X is Uniform(0,1) and Y = -ln(X), then Y ranges from 0 to infinity, not 0 to 1.

Using the wrong tail in survival function problems

S(x) = P(X > x) = 1 - F(x). When a problem asks for the probability of exceeding a deductible, you need S(d), not F(d). Mixing these up gives the complement of the correct answer.

Misapplying the law of total probability

When conditioning on a continuous random variable, you must integrate over the conditioning variable. Candidates sometimes treat a continuous conditioning variable as if it were discrete, summing instead of integrating.

Forgetting the covariance term in Var(X+Y)

Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y). The formula without the covariance term only applies when X and Y are independent. Many problems intentionally provide dependent variables to test this.

Confusing E[X^2] with (E[X])^2

The second moment E[X^2] is not the square of the mean. Var(X) = E[X^2] - (E[X])^2, so E[X^2] = Var(X) + (E[X])^2. Getting this relationship wrong leads to incorrect variance calculations.

Not conditioning properly in Bayesian problems

When the problem describes a two-stage process (e.g., first select a risk class, then observe claims), you need Bayes theorem. Candidates sometimes compute the unconditional probability when the posterior is required.

Overlooking the memoryless property of the Exponential

If X is Exponential, then P(X > s+t | X > s) = P(X > t). Failing to use this property on problems designed for it wastes several minutes doing unnecessary integration.

Errors with order statistics

The CDF of the maximum of n i.i.d. variables is F(x)^n, not n*F(x). The CDF of the minimum is 1 - (1-F(x))^n. Mixing up these formulas or applying them to non-i.i.d. variables produces wrong answers.

Arithmetic errors from skipping steps

Under time pressure, candidates try to do multi-step calculations in their heads. Write out each step on scratch paper, especially for problems involving conditional expectations or double integrals.

Common P Mistakes