1. ## Distribution Functions

Let X and Y be independent and identicall distributed random variables. Denote their common probability distribution function by F and denote their common density function by f. If

V = max{X,Y} and U = min{X,Y}

1.Express the distribution function of V in terms of F and f.

Hint: Begin with Fv(v) = P[V<=v]

2. Express the distribution function of U in terms of F and f.

2. Note that if V < v, both X and Y must be less than v.

$F_v(v) = P(V \leq v) = P(X \leq v, Y \leq v) = P(X \leq v) P(Y \leq v) =\left[ F(v)\right]^2$

Do the same for U. (begin with P(U > u))

3. Thanks, for that

so does that mean that its
F(u) = P(U>u) = P(X>u, Y>u) = P(X>u)P(Y>u) = [f(u)]^2

4. No. The cumulative distribution function $F(u)$ is defined as $P(U \leq u)$.
Note that, since P is a probability, $P(U > u) + P(U \leq u) = 1$.

See you!