Exercises
=========


#. 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.

#. Prove that $\E(aX+b) = a\E(X)+b$ for a discrete random variable.

#. Prove that $\Var(aX + b) = a^2\,\Var(X)$.

#. Prove that for $X\sim \text{Bernouilly}(p)$, the variance equals
   $\Var(X)=p(1-p)$.

#. Prove that for the Binomial Distribution the expectation $\E(X)=n
   p$.

#. Prove that for the Binomial Distribution the variance
   $\Var(X)=np(1-p)$.

#. What is the expectation of a continuous uniform distribution
   $\Uniform(a,b)$. Als give a proof.

#. What is the variance of continuous uniform distribution
   $\Uniform(a,b)$. Als give a proof.


#. 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 $R=\min(X,Y)$.