Probability: Theory & Applications

What is the definition of a sample space in probability theory?

The sample space, often denoted by S or Ω, is the set of all possible outcomes of a random experiment.

State the second axiom of Kolmogorov regarding the probability of the entire sample space.

The probability of the entire sample space S is equal to 1, denoted as P(S) = 1.

What is the addition rule for two mutually exclusive events A and B?

For mutually exclusive events, the probability of A or B occurring is the sum of their individual probabilities: P(A ∪ B) = P(A) + P(B).

Define the mathematical condition for two events A and B to be considered independent.

Two events are independent if the probability of both occurring is the product of their individual probabilities: P(A ∩ B) = P(A) * P(B).

What is the expected value (mean) of a Bernoulli random variable with success probability p?

The expected value E[X] of a Bernoulli random variable is p.

What is the general formula for the variance of a random variable X in terms of its moments?

Var(X) = E[X^2] - (E[X])^2.

State the formula for conditional probability P(A|B).

P(A|B) = P(A ∩ B) / P(B), provided that P(B) > 0.

Express Bayes' Theorem for updating the probability of a hypothesis A given evidence B.

P(A|B) = [P(B|A) * P(A)] / P(B).

What are the parameters and the Probability Mass Function (PMF) of a Binomial distribution?

Parameters are n (trials) and p (success probability). PMF: P(X=k) = (n choose k) * p^k * (1-p)^(n-k).

Under what conditions is the Poisson distribution typically used to approximate a Binomial distribution?

It is used when the number of trials n is large and the probability of success p is small, such that λ = np remains constant.