11 questions linked to/from Expected Value of Max of IID Variables
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### What is the average maximum value of a set of random numbers? [duplicate]

Let $a_1, a_2, a_3, \ldots, a_{10}$ be ten randomly chosen real numbers in the interval [0,1]. Let $m$ be the maximal value out of these 10 numbers. What is the expected value of $m$? (i.e. If i ...
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### Random Ants Question from Interview Book: How to Compute Expectation of Maximum of Random Varibales [duplicate]

I know the following question has been asked by the following link: Random ants probability question But my question is more about computing the final result: expected value of the maximum of 500 IID ...
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### Expected value of maximum of two random variables from uniform distribution

If I have two variables $X$ and $Y$ which randomly take on values uniformly from the range $[a,b]$ (all values equally probable), what is the expected value for $\max(X,Y)$?
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### Expected value of the maximum and minimum of $n$ independent random vairables

In my statistics book I was encountered by the task to calculate the maximum and minimum of $n$ independent random variables, which are uniformly distributed over $(0,1)$. Knowing the definition of ...
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### Expected falling time of all $500$ random ants

The random ant question is asked in this post. I reproduce it below for completeness. Question: $500$ ants are randomly put on a 1-foot string (independent uniform distribution for each ant ...
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### Expected Value of max + min of N d20 dice

Let's say I roll N twenty sided dice. What is the expected value of the highest and lowest rolls? Bonus: Is there a more general formula if I wanted to find the expected value of N dX dice, where ...
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### Distribution of interval that contains N independent variables

N independent random variables are uniformly distributed on [0,1]. What is the distribution function of length of minimal interval that contains all of them?
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### Finding the expectation of $\hat \theta$, if $\theta = \max(X_1, …, X_n)$ [duplicate]

I am trying to make $\hat \theta$ an unbiased estimator for a uniform distribution on $(0, \theta)$. So I am given the fact that $\hat \theta$ is the MLE for $\theta$, but I need to show that if I ...
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### Show this is an unbiased estimator for $M$

We have a sample $X_1, ..., X_n$ from $U[M,2M]$, $M>0$. Define $U=\frac{(n+1)}{(2n+1)}\max(X_i)$. I wish to show that $U$ is an unbiased estimator for $M$, but I am getting a wrong result. This is ...
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### show $\hat{\theta}$ = $\frac{x_{(n)}}{3}$ is biased

The uniform (0,3θ) distribution has pdf $f\left(x\right)=\frac{1}{3θ}$ when $0\le x\le 3θ$ and $0$ otherwise where $θ>0$ let $x_1,....x_n$ be a random sample from this population distribution ...
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### Expected value of $k$-th minimum among $n$ random variables uniformly distributed in $[0, 1]$.

Let $X_1, X_2, \ldots, X_n$ be $n$ random variables uniformly distributed in $[0, 1].$ I need to find out the expected value of $k$-th minimum. I tried finding out the cumulative distribution function ...