Exercises

1. Prove that in general \(\P(A\cup B) = \P(A) + \P(B) - \P(A\cap B)\) using only the three axioms. Hint: first write \(A\cup B\) as the union of two disjunct sets.
2. Prove that \(\E(aX+b) = a\E(X)+b\) for a discrete random variable.
3. Prove that \(\Var(aX + b) = a^2\,\Var(X)\).
4. Prove that for \(X\sim \text{Bernouilly}(p)\), the variance equals \(\Var(X)=p(1-p)\).
5. Prove that for the Binomial Distribution the expectation \(\E(X)=n p\).
6. Prove that for the Binomial Distribution the variance \(\Var(X)=np(1-p)\).
7. What is the expectation of a continuous uniform distribution \(\Uniform(a,b)\). Als give a proof.
8. What is the variance of continuous uniform distribution \(\Uniform(a,b)\). Als give a proof.