Why does this assumption give an under approximation for both the expected maximum and minimum

Question

The well known result, $\mathbb{E}[\text{min} \{X_i \}_{i=1}^n ] = \frac{1}{n+1}$ and $\mathbb{E}[\text{max} \{X_i \}_{i=1}^n ] = \frac{n}{n+1}$ where $X_i$ are I.I.D Uniform$(0,1)$ random variables is a lifesaver. It can be extrapolated to find the expected minimum and maximum of $n$ Uniform$(a,b)$ variables.

I tried to use it in the discrete case. For example, consider the maximum of $10$ rolls of a $100$ sided die, the quick formula gives: $ \frac{10}{11}\cdot 100 = 90\frac{1}{11} \approx 90.91$. The true answer however is $91.4007585757 $ (using tail sum formula : $100 - \sum\limits_{i=1}^{99}(\frac{i}{100})^{10}$ )

Notice the approximation gives an underestimate.

Now consider using it to approximate the expected minimum.

$\frac{1}{11}\cdot 100 = 9\frac{1}{11} \approx 9.09$. The true answer however is $9.59924142434$ and again this formula under-approximates?

Why does this happen in both cases? Is there a cheeky way to tweek it for the discrete case to get a bit more accurate?

Thanks!

A little thing I have noticed that might help is the following:

The approximate min + approximate max is always $n$. However, by symmetry the true min + true max is always $n+1$. (because they are centred around the mean of $\frac{n+1}{2}$

Answer 1

It under-approximates because you are approximating a random variable taking values in $$ \left\{\frac{1}{k},\frac{2}{k},\dots,\frac{k-1}{k}, 1\right\} $$ by a uniform random variable, that is, uniform on $$ \left(0,\frac{1}{k}\right] \cup \left(\frac{1}{k}, \frac{2}{k}\right]\cup \dots\cup \left(\frac{k-1}{k}, 1\right]\,. $$ Viewed differently, you replace the value $$ \frac{i}{k} $$ (for $1\leq i\leq k$) by a uniformly random value in $$ \left(\frac{i-1}{k}, \frac{i}{k}\right] $$ which leads to a (small) under-approximation.

Answer 2

Note that for $X$ positive we have $\mathbb EX=\int_0^\infty 1-F(x)\,\mathrm d x$. Now note that if $X$ is the discrete variant and $Y$ is the continuous approximation. Then $F_X\leq F_Y$ (as $F_X(x)=F_Y(x)$ for $1\leq x\leq 100$ and $F_Y$ is linear while $F_X$ is constant on $[x,x+1)$.).

Thus this also holds true for the distribution of max and min, and thus by above formula the expected values for $Y$ are smaller than those for $X$.