Let T1, T2, · · · , Tn be independent and identically distributed random variables with common c.d.f. F(t)

Question

Let T1, T2, · · · , Tn be independent and identically distributed random variables with common c.d.f. F(t).

a- Let Mn = max(T1, T2, · · · , Tn). Find the c.d.f. of Mn in terms of F. (Hint: Note that P(Mn ≤
t) = P(T1 ≤ t, T2 ≤ t, · · · , Tn ≤ t) and use independence of Ti
’s.)

b- Let Nn = min(T1, T2, · · · , Tn). Find the c.d.f. of Nn in terms of F.

c- Suppose that each Ti has Weibull distribution with parameters α > 0 and β > 0. Find E(Nn).*

I cant figure out how to solve it.

Please suggest me a way or tell me what to do?

Answer 1

Here are a few hints:

1) If $X$ and $Y$ are independent random variables, with CDF $F_{X}(x)$ and $F_{Y}(y)$, respectively, then $P(X\leq x, Y \leq y)=P(X\leq x\cap Y \leq y)=P(X\leq x) P(Y \leq y)=F_{X}(x) F_{Y}(y)$.

2) If $X_{1}$ and $X_{2}$ are identically distributed, then $F_{X_{1}}(x)=F_{X_{2}}(x)\equiv F_{X}(x)$.

3) Taking the random variable $M_{n}$, as defined above, the event $\lbrace M_{n} \leq t \rbrace$ is equivalent to the event $\lbrace T_{1}\leq t \cap T_{2} \leq t,...,\cap T_{n}\leq t \rbrace$.

4) Taking the random variable $N_{n}$, as defined above, the event $\lbrace N_{n} > t \rbrace$ is equivalent to the event $\lbrace T_{1}>t \cap T_{2} >t,...,\cap T_{n}>t \rbrace$.