# Maximum of RVS independent and identically distributed

I am having a small doubt regarding maximum of random variables. I have $$Z= \max\{ X_1, X_2,\dots X_p, \dots X_N\}$$ where all $X_i$ are independent, identically distributed. Now, If for sure, I know that $$\{X_1,\dots,X_p\}>X_{p+1}>\dots>X_N$$ the can i say that the CDF of $Z$,

\begin{align*}P(Z<z)&= P\left(\max\{ X_1, X_2,\dots X_p, \dots X_N\}<z\right)\\&=P\left(\max\{ X_1, X_2,\dots X_p\}<z\right)\\&=\left[P(X_1<z)\right]^{N-p-1}\end{align*}

How to find CDF of $Z$ in this case ? Is there any theorem that reduce the size of total variables for certain condition ?

If you can say for sure that the event $$E:=\left\lbrace\{X_1,\dots,X_p\}>X_{p+1}>\dots>X_N\right\rbrace$$ occurs, then you can express this as $$P(E)=P\left(\{X_1,\dots,X_p\}>X_{p+1}>\dots>X_N\right)=1$$ Therefore the distribution of $Z$ given this information can be expressed as \begin{align*}F_{Z\mid E}(z|E)&=\frac{P(Z<z, E)}{P(E)}\overset{P(E)=1}=\\&=P\left(\max\{X_1, X_2, \ldots X_N\}<z \, \text { and } \, \left\lbrace\{X_1,\dots,X_p\}>X_{p+1}>\dots>X_N\right\rbrace\right)\\&= P\left(\max\{X_1, X_2, \ldots X_p\}<z\right)=\ldots\\&=\left[F_{X_1}(z)\right]^{N-p-1}\end{align*} as you have it. So, concerning your question, the notion of the conditional distribution is what you need in order to express your correct conclusion.