Random variables, moment generating functions, limit theorems, Markov chains, Poisson processes and martingales.
40 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a random variable?
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A function mapping from a sample space to the real numbers- Define the cumulative distribution function (CDF) of a random variable X
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F(x) = P(X <= x), the probability that X is at most x- What is the difference between a probability mass function and a probability density function?
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PMF applies to discrete RVs and gives probabilities directly; PDF applies to continuous RVs and must be integrated to give probabilities- State the definition of expected value (mean) of a discrete random variable X
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E[X] = sum over x of x * P(X = x)- Define variance of a random variable X
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Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2- What is covariance between two random variables X and Y?
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Cov(X, Y) = E[(X - E[X])(Y - E[Y])], measuring how X and Y move together- When are two random variables X and Y independent?
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When P(X <= x, Y <= y) = P(X <= x) * P(Y <= y) for all x and y- What is a moment generating function (MGF)?
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M_X(t) = E[e^(tX)], a function whose derivatives at zero yield the moments of X- What is the MGF of a standard normal distribution N(0, 1)?
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M(t) = e^(t^2/2)- What is the MGF of an exponential distribution with rate lambda?
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M(t) = lambda / (lambda - t) for t < lambda- State the uniqueness theorem for MGFs
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If two random variables have the same MGF in a neighborhood of zero, they have the same distribution- What is the relationship between the MGF M(t) and the moments of a random variable?
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The kth moment equals the kth derivative of M(t) evaluated at t = 0: E[X^k] = M^(k)(0)- If X and Y are independent random variables, what is the MGF of X + Y?
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M_(X+Y)(t) = M_X(t) * M_Y(t)- State the Central Limit Theorem
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If X1, X2, ... are iid with mean mu and variance sigma^2, then (sum X_i - n*mu) / (sigma * sqrt(n)) converges in distribution to N(0, 1)- State the Weak Law of Large Numbers
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If X1, X2, ... are iid with mean mu, then the sample mean converges in probability to mu
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