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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$𝑋_k$≀π‘₯)=𝑃(𝑋1 ≀ π‘₯ ∩ 𝑋2 ≀ π‘₯ ∩ ⋯𝑃 𝑋n ≀ π‘₯) explain why

OR

=𝑃(𝑋1≀π‘₯)𝑃(𝑋2≀π‘₯)β‹― 𝑃(𝑋 ≀π‘₯) 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: $𝑓(π‘₯) = \frac{1}\theta{} \ \ \ \ (0 < π‘₯ < πœƒ)$

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  • $\begingroup$ The maximum of the observations is a good estimator of ΞΈ, It's the MLE. But it is obviously biased: The max must always be smaller than ΞΈ. 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 $\endgroup$
    – BruceET
    Oct 31, 2017 at 16:46
  • $\begingroup$ @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. $\endgroup$
    – localplutonium
    Oct 31, 2017 at 17:53
  • $\begingroup$ 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 $X_{(i+1)}.$ $\endgroup$
    – BruceET
    Oct 31, 2017 at 21:09

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