1. ## Standard Uniform Distribution

Never really did much stats so this is probably an easy question.

What values can the standard uniform distribution take? Is it just any value between 0 and 1? Is each value chosen with equal probability?

So for example, if I'm to asked the question...

Calculate $t_1$ ~ $U(0,1)$ where U is the uniform distribution and t is a random variable. Is that even a properly written out question? Does it just mean t will take a random value between 0 and 1?

As I say I've not done much stats and not entirely sure what the notation stands for.

Calculate $t_1$ ~ $U(0,1)$ where U is the uniform distribution and t is a random variable. Is that even a properly written out question? Does it just mean t will take a random value between 0 and 1?
I would say that is a good guess.
Only your textbook can say for sure.

Never really did much stats so this is probably an easy question.

What values can the standard uniform distribution take? Is it just any value between 0 and 1? Is each value chosen with equal probability?

So for example, if I'm to asked the question...

Calculate $t_1$ ~ $U(0,1)$ where U is the uniform distribution and t is a random variable. Is that even a properly written out question? Does it just mean t will take a random value between 0 and 1?

As I say I've not done much stats and not entirely sure what the notation stands for.

$t$ ~ $U(0,1)$ means that t has a density that is 1 on (0,1) and 0 otherwise.

4. Hmm I'm still not totally clear on this... Let me put down whats in the textbook. Its to do with the Kolmogorov-Smirnov distribution.

I'm testing the hypothesis that a sample of n independent observations comes from a specified continuous distribution.
...Let $F_0(x)$ be the distribution function from which, according to the hypothesis to be tested, the sample has been taken. We let $x_1, x_2, \ldots , x_n$ denote the order statistics of the sample, and define $t_j = F_0(x_j)$ for $j = 1,2, \ldots , n$.
Then I have a formula to use.
If the null hypothesis holds $t_1, t_2, \ldots t_n$ are the order statistics of a random sample size n from the uniform distribution on (0,1).

And the first thing I have written down to do (written by my prof)...
Set n to be a fixed amount
Calculate $t_1, t_2, \ldots , t_n$ ~ U(0,1)

note, $F_0(x)$ is not given but i assume thats the empirical distribution function?

So, given this info, how would you calculate the t's for say, n=4?

As I say im quite new to stats and this is all the info I have. If I can calculate the t's I'm sorted for the rest of it.