Past paper quick fire questions (answer with what method or formula to use ) Flashcards
(12 cards)
Q: How to show that P(E∩F∩G) = P(E)P(F)P(G) and determine if E, F, and G are independent?
A: Use the multiplication rule for probabilities and check if P(E∩F∩G) = P(E)P(F)P(G). E, F, and G are independent if their pairwise probabilities factorize.
Q: How to compute the expectation and variance for a discrete random variable X with given probabilities?
A: Use the formulas: E(X) = Σ [x * P(x)] and Var(X) = Σ [ (x - E(X))^2 * P(x) ]. Also, compute E(2X+3) and Var(2X+3) using linear transformations.
Q: How to compute conditional probability?
A: Use the formula P(A|B) = P(A ∩ B) / P(B). For example, compute P(X = 1 | X ∈ A) by finding the joint and marginal probabilities.
Q: How to compute the joint distribution for random variables X and Y?
A: Use the provided joint probabilities in the table. Compute the marginals and verify any calculations using the known properties of joint distributions.
Q: How to calculate the probability of an event using joint distributions?
A: Sum the appropriate joint probabilities for the event of interest. For example, P(X ∈ A) = Σ P(X, Y) for all values of Y in A.
Q: How to verify independence of random variables X1 and X2?
A: Check if the joint probability equals the product of the marginal probabilities: P(X1, X2) = P(X1) * P(X2). If true, they are independent.
Q: How to find the conditional distribution of X given Y?
A: Use the formula P(X|Y) = P(X,Y) / P(Y). Verify the result by checking that the sum of the conditional probabilities equals 1.
Q: How to compute the marginal probability for a discrete random variable?
A: Sum the joint probabilities over the range of the other random variable: P(X) = Σ P(X, Y).
Q: How to check if two random variables are uncorrelated?
A: Compute the covariance: Cov(X, Y) = E[XY] - E[X]E[Y]. If Cov(X, Y) = 0, then the random variables are uncorrelated.
Q: How to compute the expectation for a random variable given a conditional distribution?
A: Use the formula E(X|Y) = Σ [x * P(X|Y)] and compute the expected value based on the conditional distribution given Y.
Q: How to compute the joint pmf and marginal pmf?
Joint pmf = P(X,Y) = P(X|Y)P(Y)
Marginal pmf = Sum of joint pmf over all possible values of Y :
P(X) = Sum (values of Y) of P(X,Y)
Q: How to calculate the mean and variance of Y = max(X1, X2)?
A: Use the cumulative distribution function (CDF) of the maximum to determine its distribution, then calculate E[Y] and Var[Y].
