# What makes the cdf curve of a uniform distribution a constant positive slope? [duplicate]

Assignment asks to explain why max of X will have the highest probability..

What is the reason why the probabilities of each random variable create a constant positive slope?

𝑃(max$𝑋_k$≤𝑥)=𝑃(𝑋1 ≤ 𝑥 ∩ 𝑋2 ≤ 𝑥 ∩ ⋯𝑃 𝑋n ≤ 𝑥) explain why

OR

=𝑃(𝑋1≤𝑥)𝑃(𝑋2≤𝑥)⋯ 𝑃(𝑋 ≤𝑥) explain why

is this because of linearity and independence ?

This is the very first statistics class I've taken. I'm still learning.

I am suppose to find an unbiased estimator for a given pmf for a uniform distribution.

pmf: $𝑓(𝑥) = \frac{1}\theta{} \ \ \ \ (0 < 𝑥 < 𝜃)$

• The maximum of the observations is a good estimator of θ, It's the MLE. But it is obviously biased: The max must always be smaller than θ. A constant multiplier (depending on n) of the max can be used to get an unbiased estimator Let $X_{(n)}$ denote the max, Find its dist'n and then find $E[X_{(n)}].$. – BruceET 9 mins ago Oct 31, 2017 at 16:46
• @BruceET semi duplicate more trying to figure out why the independent probabilities are ascending order. Just need an answer to the assignment question "explain why" ... I've proved / did the rest of showing the estimator to be unbiased. (n+1)/n *maxX was given. Oct 31, 2017 at 17:53
• The original observations are random. They are used to get the distribution of the max. Of course the sorted observations are not independent because the $i$th order statistic $X_{(i)}$ is smaller than the $(i+1)$st order statistic $X_{(i+1)}.$ Oct 31, 2017 at 21:09