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Edit: A lot of my computations in the question below are utterly wrong, I just left them as they since the comments/answers applied to the original form.

Edit: I have updated the question to hopefully better explain the draws I am referring to. In the actual problem, $x_{(i)}, i\in\{1,\ldots,n\}$ are the $n$ lowest out of $m$ i.i.d. draws from $U[0,1]$. I hope I am making a duly simplification below.

Let $x_{(1)}\leq\ldots\leq x_{(n)}$ be $n$ i.i.d. draws of a random variable uniformly distributed in $[0,1]$. I know that (if I am not mistaken)

\begin{alignat*}{2} E[x_{(i)}]&=\frac{i}{n+1}\\ E[\sum_{i=1}^n x_{(i)}] &=\frac{n}{2}, \\ E[\sum_{i=1}^n (x_{(i)} ^2)] &= \sum_{i=1}^n\left(\frac{i}{n+1}\right)^2=\frac{n(2n+1)}{6(n+1)}. \end{alignat*}

I furthermore know that \begin{align} \underbrace{E[x_{(i)}^2]}_{\frac{2n+1}{6(n+1)}}\neq\underbrace{(E[x_{(i)}])^2}_{\frac{1}{4}} \end{align} and hence was expecting that \begin{align} E[\left(\sum_{i=1}^n x_{(i)}\right)^2]\neq\left(E[\sum_{i=1}^n x_{(i)}]\right)^2. \end{align} However, my calculation is \begin{alignat*}{3} E[\left(\sum_{i=1}^n x_{(i)}\right)^2]% &= E[\sum_{i=1}^n (x_{(i)}^2)+2\sum_{i=1}^n x_{(i)} \sum_{j=1}^{i-1}x_{(j)}] \\ &= \frac{n(2n+1)}{6(n+1)} + 2\sum_{i=1}^n \frac{i}{n+1} \frac{(i-1)i}{2(n+1)}\\ &= \frac{n(2n+1)}{6(n+1)} + \frac{1}{(n+1)^2}\sum_{i=1}^n (i^3-i^2)\\ &= \frac{n(2n+1)}{6(n+1)} + \frac{1}{(n+1)^2}\left(\left(\frac{n(n+1)}{2}\right)^2-\frac{n(n+1)(2n+1)}{6}\right)\\ &= (\ldots) = \left(\frac{n}{2}\right)^2=\left(E[\sum_{i=1}^n x_{(i)}]\right)^2 \end{alignat*}

Are the two (i.e. expected value of squared sum, and square of expected sum) really equal or have I made a mistake in my calculations? If they are equal, is there an intuition why squares and expectations are interchangeable here but not in the case above (i.e. for the sum but not for the individual $x_{(i)}$)?

(Apologies if the calculations are somewhat cumbersome, I have tried to simplify the original problem with more parameters to this form.)

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    $\begingroup$ In the second line why $\displaystyle E[X_i^2] =\left(\frac {i} {n+1}\right)^2$? If $X_i$ are i.i.d. then $E[X_i]$ should be the same and should be independent of the dummy variable $i$. $\endgroup$ – BGM Sep 28 '16 at 15:00
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    $\begingroup$ Note that $\mathbb{E}(Y^2) = (\mathbb{E}(Y))^2$ if, and only if, the variance of $Y$ equals $0$. If you apply this for $Y := \sum_{i=1}^n X_i$, then this means that the two expressions $$\mathbb{E} \left[ \left( \sum_{i=1}^n X_i \right)^2 \right] \quad \text{and} \quad \left[ \mathbb{E} \left( \sum_{i=1}^n X_i \right) \right]^2$$ are not equal. $\endgroup$ – saz Sep 28 '16 at 15:15
  • $\begingroup$ @BGM I fear I misphrased my question (or over-simplified it). Originally, the $n$ draws are the highest out of $m$, so I am really looking at $x_{(1)},\ldots,x_{(n)}$. I'll rephrase my question accordingly and hope it makes more sense, and that at least my first calculations are still correct. $\endgroup$ – Bernd Sep 30 '16 at 12:21
  • $\begingroup$ I deleted my answer. I thought it was easy to solve with the Irwin-Hall distribution. (It solved the problem for $n$ random variables, I've just seen you've got $n$ out of $m$ so that completely changes the question and I'm not sure my answer is valid. $\endgroup$ – Joaquin San Oct 10 '16 at 18:53
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Edit I will consider the total cases are $n$. There were many mistakes in my last answer so I'm doing it a little bit different.

Let's see a fact: $ \displaystyle\sum_{i=1}^n x_{(i)} = \sum_{i=1}^n x_i $ because for the sum, the order doesn't matter. And therefore, as every $x_{(i)}$ is uniform in $(0,1)$, our sum $Y = \displaystyle\sum_{i=1}^n x_{i}$ has an Irwin-Hall distribution with parameter $n$.

Therefore $ \mathbf{E} \left[ \displaystyle\left(\sum_{i=1}^n x_i\right)^2\right] = \mathbf{E}[Y^2] = Var(Y)+\left(\mathbf{E}[Y]\right)^2 = \frac{n}{12}+\left(\frac{n}{2}\right)^2 $.

You can definetely conclude from here. I hope there aren't any dumb mistakes this time.

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    $\begingroup$ I'll edit everything and let you know. $\endgroup$ – Joaquin San Sep 30 '16 at 12:38
  • $\begingroup$ I corrected it some days ago $\endgroup$ – Joaquin San Oct 8 '16 at 15:09
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    $\begingroup$ Yes. My mistake. I'll correct it. $\endgroup$ – Joaquin San Oct 10 '16 at 13:17
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$ – Joaquin San Oct 10 '16 at 18:55
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    $\begingroup$ For the sake of completeness, the same result can be obtained the following (more cumbersome) way: $E[(\sum_{i=1}^n x_{(i)})^2]=E[\sum_{i=1}^n (x_{(i)}^2)]+2\sum_{i=1}^n\sum_{j=1}^{i-1}E[ x_{(i)}\cdot x_{(j)}]$, where the first summand equals $\frac{n}{3}$ and for the second (where I previously wrongly assumed that $x_{(i)}, x_{(j)}$ were independent), I used this answer (math.stackexchange.com/a/400742/83069). It yields $\frac{n(1+3n}{12}$ as in Joaquin's answer. $\endgroup$ – Bernd Oct 12 '16 at 7:02

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