2.12. 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.

  9. Let \(X\) and \(Y\) be two independent continuous random variables with pdf \(f_X\) and \(f_Y\) respectively.

    1. We consider the calculated random variable \(Q=\max(X,Y)\). Prove that \(f_Q(x) = F_X(x)f_Y(x)+f_X(x)F_Y(x)\). Hint: start by calculating \(F_Q(x,y)\) and observing that \(F_Q(x)=\P(Q\leq x)=\P(X\leq x \text{ and } Y\leq x)\).

    2. Also give expressions for \(f_R\) and \(F_R\) for the random variable \(R=\min(X,Y)\).