Cfrgtkky

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?

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 $P(Xleq x, Y leq y)=P(Xleq xcap Y leq y)=P(Xleq x) P(Y leq y)=F_{X}(x) F_{Y}(y)$.

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 $lbrace T_{1}>t cap T_{2} >t,...,cap T_{n}>t rbrace$.