The mechanics of solving the linear system were fine but the uniqueness part tripped me up a bit.
First, explain that the stationary distribution is the left eigenvector of P corresponding to eigenvalue 1, and that it can be found by solving the linear system π(P - I) = 0 with the normalization constraint. Then, discuss why irreducibility and aperiodicity guarantee a unique stationary distribution, and outline the computational steps to solve the system.
Pro tip: Mention that in practice, for large state spaces, iterative methods like power iteration are used instead of direct linear algebra, and note that the stationary distribution is also the limiting distribution due to aperiodicity and irreducibility.
Write the stationarity condition πP = π as a system of linear equations, and include the normalization constraint ∑π_i = 1.
Solve the system by replacing one of the redundant equations with the normalization constraint, then use standard linear algebra techniques (e.g., Gaussian elimination) to find π.
Argue that irreducibility ensures a unique stationary distribution (up to scaling), and aperiodicity ensures that the chain converges to it, so the normalized solution is unique.
Mention that for large chains, direct methods may be inefficient, and iterative methods like power iteration or solving the eigenvector problem numerically are preferred.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clearly stating the definitions of the Exponential and Poisson distributions and their respective probability mass/density functions. Then, derive E[X] by setting up and evaluating the integral ∫₀^∞ x λ e^{-λx} dx using integration by parts, and derive E[N] by evaluating the sum ∑_{n=0}^∞ n e^{-λ} λ^n / n! by simplifying the term and recognizing the series expansion of e^λ. Conclude by noting that both expectations equal 1/λ and λ respectively, and briefly comment on the relationship between the two distributions.
Pro tip: Emphasize the connection between the Exponential and Poisson distributions—both arise from a Poisson process—and mention that the rate parameter λ represents the average number of events per unit time, so the expected waiting time between events is 1/λ. This shows deeper understanding beyond rote calculation.
Clearly write the PDF of X ~ Exponential(λ) as f(x) = λ e^{-λx} for x ≥ 0, and the PMF of N ~ Poisson(λ) as P(N = n) = e^{-λ} λ^n / n! for n = 0, 1, 2, ... . State that you will compute E[X] = ∫₀^∞ x f(x) dx and E[N] = ∑_{n=0}^∞ n P(N = n).
Set up the integral E[X] = ∫₀^∞ x λ e^{-λx} dx. Use integration by parts with u = x, dv = λ e^{-λx} dx, so du = dx and v = -e^{-λx}. Evaluate the boundary term [ -x e^{-λx} ]₀^∞ = 0 and the remaining integral ∫₀^∞ e^{-λx} dx = 1/λ, giving E[X] = 1/λ.
Write E[N] = ∑_{n=0}^∞ n e^{-λ} λ^n / n!. Note that the n=0 term is zero, so start from n=1. Cancel n with n! to get n/n! = 1/(n-1)!, factor out λ e^{-λ}, and re-index the sum to obtain λ e^{-λ} ∑_{k=0}^∞ λ^k / k! = λ e^{-λ} e^{λ} = λ.
State the final results: E[X] = 1/λ and E[N] = λ. Briefly explain the intuitive link: in a Poisson process with rate λ, the expected number of events in a unit interval is λ, while the expected waiting time between events is 1/λ.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, explain that a mixed-strategy Nash equilibrium in a 2x2 zero-sum game can be found by making each player indifferent between their two pure strategies. Then, set up equations for Player A's expected payoff from each action (given Player B's probabilities) and for Player B's expected payoff from each action (given Player A's probabilities), solve for the probabilities, and verify the equilibrium.
Pro tip: Mention that the equilibrium probabilities depend only on the payoffs of the opponent, not on your own payoffs, and that you can quickly check your work by ensuring the expected payoffs for each player are equal across their actions.
Label the payoff matrix entries (a, b, c, d) for Player A, and let p be the probability Player A chooses the first action, and q be the probability Player B chooses the first action.
For Player A, set expected payoff from action 1 equal to expected payoff from action 2, given q. For Player B, set expected payoff from action 1 equal to expected payoff from action 2, given p.
Solve the two linear equations for p and q. The solution gives the mixed-strategy Nash equilibrium probabilities.
Check that p and q are between 0 and 1. If not, the equilibrium may be pure. Also, compute expected payoffs to ensure they are equal across actions for each player.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.