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I was looking at this post because there was something similar in my school's lecture slide Convergence in Probability.

However in my school's slide it didn't mention how it got the cdf so I have 2 questions.

What does it mean to take the maximum of a set of distributions? My understanding is that the distribution $X_n$ is just a set of data that generates a pattern that can be modeled/estimated by a distribution, so what does $\max\{X_1,...,X_n\}$ mean?

How do you get $F_{Y_n}(t) = \frac{t^n}{\theta^n}$, since it's a uniform distribution the density function would be $f_y(t) = \frac{1}{\theta}$ but integrating $f_y(t)$ between $0$ to $t$ does not give me $F_{Y_n}(t)$.

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2 Answers 2

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The $X_i$ don't denote "distributions" but random variables.

Let's do an example. Throw a fair die (independently) $n$ times. Let $X_i$ be the value of the $i$-th throw. An experiment might give (for $n=5$) $X_1=3$, $X_2=5$, $X_3=3$, $X_4=5$ and $X_5=2$. In this case $\max(X_1,\ldots,X_5)=5$ for this outcome. So $\max(X_1,\ldots,X_5)$ is a random variable. It takes values in $\{1,2,3,4,5,6\}$ like the $X_i$ do, but it is more likely to take larger values than smaller ones.

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  • $\begingroup$ thank you very much! your explanation helped alot but I can only choose one accepted answer $\endgroup$
    – demalegabi
    Commented May 7, 2017 at 12:15
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This is a good question. To get some intuition as to what $\max\{X_1,...,X_n\}$ means lets look at what we are assigning this quantity to: $$Y_n=\max\{X_1,...,X_N\}$$ $Y_n$ is a random variable, so all we are doing is looking at a function of a set of random variables. We aren't picking one out and saying this one is the max, we can only do that once we have a realization of the data. Instead, we are just writing down a symbol for the concept of the max of a random sample so that we can do algebra on it.

For your second question: you are basically already there.

What is $P(Y_n <y)$? Well if the max is less than $y$ then all $X_i$ are less than $y$. If those $X_i$ are i.i.d then the product of the joint distribution $$P(X_1 <y,X_2<y,...,X_n<y)$$

factors into the product of the marginals $$\prod_{i=1}^nP(X_i<y)$$ and you already know the formula for the marginal density function, so find the marginal cdf and plug-in!

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