I have troubles solving this task:

Let $U_1,U_2,\dots$ be an i.i.d. sequence of random variables with uniform distribution on $[0,1]$. We set for every integer $n\geq 1$ $$M_n=\max\{1/\sqrt{U_1},\dots,1/\sqrt{U_n}\}$$

Compute the distribution function $F_n(x)=P(M_n\leq x)\quad \text{for } x\geq 0$

What I have so far:

Since $U_i$ are i.i.d. we have $$P(M_n\leq x)=P(1/\sqrt{U_1}\leq x)^n=P(\frac{1}{x^2}\leq U_1)^n=\int_{x^{-2}}^\infty \mathbb{1}_{[0,1]}\mathrm{d}x$$ but here I get stucked. I don't know what to do. I think I have to do a substitution but this confuses me since I do not have a variable in the integrand?

  • $\begingroup$ You should not integrate with respect to $x$. Integrate with respect to another variable. $\endgroup$ – Ken Duna Jun 30 '16 at 14:36

The error comes right at the last step.

$$P\left(\frac{1}{x^2}\leq U_1\right)^n = \left(\int_{\frac{1}{x^2}}^1 dy\right)^n = \left(1-\frac{1}{x^2}\right)^n$$

Note that this is for $x \geq 1$.

For $x<1$, $P(M_n\leq x) = 0$.

  • $\begingroup$ Hi, thanks a lot for the answer. Now it's clear. $\endgroup$ – MarcE Jun 30 '16 at 15:16

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