I am studying probability theory and I need to understand moment generating functions (MGFs). The MGF of a random variable X is: M_X(t) = E[e^(tX)] = ∑_x e^(tx) P(X = x) & discrete \ ∫_(-∞)^(∞) e^(tx) f_X(x) dx & continuous My questions: 1. Why is it called a "moment generating" function? How do I extract moments from it? 2. Find the MGF of X ∼ Exponential(λ) and use it to find E[X] and E[X²]. 3. What is the domain of t for which the MGF exists? 4. How do MGFs help prove the sum of independent Poissons is Poisson? 5. What is the relationship between MGFs and characteristic functions? I want to see the computations step by step.

Bayes' theorem states: P(A|B) = P(B|A) P(A)/P(B) I understand the formula, but I want to build intuition. Can someone explain: 1. Why Bayes' theorem is so important in statistics and machine learning 2. A concrete real-world example where using Bayes' theorem gives a counterintuitive result 3. The difference between the Bayesian and frequentist interpretations of probability A classic example is medical testing with rare diseases. How does the prior probability P(A) affect the posterior probability P(A|B)?