0
$\begingroup$

$X_1$, $X_2$,... are iid random variables having pdf $$f(x)=3x^2 I_{(0,1)}(x)$$

We also have that $V_n = n^{1/3}$ min$(X_1,...,X_n)$ and $W_n = n^{1/3}$ max$(X_1,...,X_n)$.

a.) Consider the sequence $V_1, V_2,...$ and give the pmf or pdf of the limiting distribution

b.) Consider the sequence $W_1, W_2,...$ and give the pmf or pdf of the limiting distribution

I know how to solve a problem like this when $X_1, X_2,...$ are uniform iid random variables, but I am having trouble figuring out how to start this problem given that the pdf is different.

For part a.) this is what I have so far: The support of $V_n$ is $(0,n^{1/3})$, $$F_{v_n}(v) = P(V_n) \leq v $$ $$=P(n^{1/3} min \{X_1,...,X_n\} \leq v/n^{1/3}) $$ $$=1-[P(X_1 > v/n^{1/3})]^n$$ $$=1-[1-P(X_1 \leq v/n^{1/3})]^n$$

I am getting stuck after this trying to determine the cdf and also don't know if this is correct.

$\endgroup$
2

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.