Independent and identically distributed random variable questions.

Question

So I am currently studying Probability and Statistics and was wondering if anyone could help me to understand this questions.

This is what I have gathered from my notes so far to aid me in answering this but I don't know if this is correct as my notes don't have any examples or text specifically relating to these sort of questions:

Let $Mn=max(X1,X2,…,Xn)$. where $X1,…,Xn$ are independent and uniformly distributed.

The distribution function of the maximum is the joint probability that:

$Xk≤x$ for all $k FM(x)=P(Mn≤x)=P(X1≤x,…,Xn≤x)=xn for 0≤x≤1$. Also $FM(x)=0 for x<0

and FM(x)=1 for x>1$.

The density function is

$fM(x)=F′M(x)=nxn−1$

So I'm not sure whether I should be using this and saying that:

$Y_3=max(X_1,X_2,X_3)$

and that I need to find the distribution function in terms of $Y$ rather than $X$

and then let $F'_Y(x)=nxn-1$

And this is where my ideas/knowledge of this question end unfortunately. Can anyone advise how to further tackle this please.

Answer 1

• $P(Y>1) = 1- P(Y\le1)$ so it suffices to compute $P(Y \le 1)$
• $$P(Y \le 1) = P(X_1 \le 1, X_2 \le 1, X_3 \le 1) = P(X_1 \le 1) P(X_2 \le 1) P(X_3 \le 1)$$ (make sure you know how to justify each equality above)
• $P(X_i \le 1) = F(1)$ for each $i$