Wednesday, December 26, 2018
Wednesday, October 31, 2018
Tuesday, October 23, 2018
The 9th Congress of Romanian Mathematicians
The 9th Congress of Romanian mathematicians will take place at Galați Romania from June 28 to July 3 2019. More details at the Congress website.
Sunday, October 21, 2018
Notes on Elementary Probability
For the first time, I have self-published with Amazon. I liked how my undergraduate probability notes turned out and I decided to publish them with Amazon. You can get them here. I will still make them available at no cost to my students in electronic form. Here is the cover.
By the way, I here is my author page on Amazon
Monday, October 1, 2018
Wednesday, June 6, 2018
About order statistics.
Suppose that $X_1,\dotsc , X_n$ are independent random variables uniformly distributed in $[0,1]$. Denote by $Y_i=X_{(i)}$ its order statistics.
The random vector $(Y_1,\dotsc, Y_n)$ has distribution
$$
p(y_1,\dotsc, y_n)=\begin{cases}
n!, & 0\leq y_1\leq \cdots \leq y_n\leq 1, \\
0, & {\rm otherwise}.
\end{cases}
$$
The random variable $Y_k$ has distribution
$$
p_k(y_k)=n!\int_{\substack{0\leq y_1\cdots \leq y_k\\ y_k\leq y_{k+1}\leq \cdots \leq y_n\leq 1}}dy_1\cdots dy_{k-1}dy_{k+1}\cdots dy_{n}
$$
$$
= n!\left(\int_{0\leq y_1\leq \cdots \leq y_{k-1} \leq y_k} dy_1\cdots dy_{k-1}\right)\left(\int_{y_k\leq y_{k+1}\leq \cdots \leq y_{n} \leq 1} dy_{k+1}\cdots dy_{n}\right)
$$
$$
= \frac{n!}{(k-1)!(n-k)!} y_k^{k-1}(1-y_k)^{n-k}
$$
Thus $p_k$ is a $B(k,n+1-k)$-distribution. $\newcommand{\bE}{\mathbb{E}}$ We have
$$
\bE[Y_k]=\frac{n!}{(k-1)!(n-k)!}\int_0^1 y^k(1-y)^{n-k} dy=\frac{n!}{(k-1)!(n-k)!}\frac{\Gamma(k+1)\Gamma(n-k+1)}{\Gamma(n+2)}
$$
$$
= \frac{n!}{(k-1)!(n-k)!}\frac{k!(n-k!)}{(n+1)!}=\frac{k}{n+1}.
$$
$$
\bE[Y_k^2]=\frac{n!}{(k-1)!(n-k)!}\int_0^1 y^{k+1}(1-y)^{n-k} dy= \frac{n!}{(k-1)!(n-k)!}\frac{\Gamma(k+2)\Gamma(n-k+1)}{\Gamma(n+3)}
$$
$$
=\frac{n!}{(k-1)!(n-k)!}\frac{(k+1)!(n-k)!}{(n+2)!}=\frac{k(k+1)}{(n+1)(n+2)}.
$$
$\DeclareMathOperator{\Var}{Var}$. Hence
$$
\Var[Y_k]= \frac{k(k+1)}{(n+1)(n+2)}-\left(\frac{k}{n+1}\right)^2=\frac{k}{n+1}\left(\frac{k+1}{n+2}-\frac{k}{n+1}\right)=\frac{k(n+1-k)}{(n+1)^2(n+2)}.
$$
We deduce
$$
\Delta_n=\sum_{k=1}^n \Var[Y_k]=\frac{1}{(n+1)^2(n+2)}\sum_{k=1}^n k(n+1-k).
$$
We have
$$
\sum_{k=1}^nk(n+1-k)= (n+1)\sum_{k=1}^n kn-\sum_{k=1}^n k(k-1)
$$
$$
= \frac{n^2(n+1)}{2}-\sum_{k=1}^nk(k-1).
$$
Now we write
$$
k(k-1)=\frac{1}{3}\Big(\; k^3-(k-1)^3-1\;\Big)
$$
so
$$
\sum_{k=1}^n k(k-1)=\frac{1}{3}\Big(\; n^3 -n\;\Big)=\frac{n(n+1)(n-1)}{3}.
$$
We deduce
$$
\Delta_n= \frac{1}{(n+1)^2(n+2)}\left(\;\frac{n^2(n+1)}{2}-\frac{n(n+1)(n-1)}{3}\;\right)=\frac{n(n+1)(n+2)}{6(n+1)^2(n+2)}=\frac{n}{6(n+1)}.
$$
The random vector $(Y_1,\dotsc, Y_n)$ has distribution
$$
p(y_1,\dotsc, y_n)=\begin{cases}
n!, & 0\leq y_1\leq \cdots \leq y_n\leq 1, \\
0, & {\rm otherwise}.
\end{cases}
$$
The random variable $Y_k$ has distribution
$$
p_k(y_k)=n!\int_{\substack{0\leq y_1\cdots \leq y_k\\ y_k\leq y_{k+1}\leq \cdots \leq y_n\leq 1}}dy_1\cdots dy_{k-1}dy_{k+1}\cdots dy_{n}
$$
$$
= n!\left(\int_{0\leq y_1\leq \cdots \leq y_{k-1} \leq y_k} dy_1\cdots dy_{k-1}\right)\left(\int_{y_k\leq y_{k+1}\leq \cdots \leq y_{n} \leq 1} dy_{k+1}\cdots dy_{n}\right)
$$
$$
= \frac{n!}{(k-1)!(n-k)!} y_k^{k-1}(1-y_k)^{n-k}
$$
Thus $p_k$ is a $B(k,n+1-k)$-distribution. $\newcommand{\bE}{\mathbb{E}}$ We have
$$
\bE[Y_k]=\frac{n!}{(k-1)!(n-k)!}\int_0^1 y^k(1-y)^{n-k} dy=\frac{n!}{(k-1)!(n-k)!}\frac{\Gamma(k+1)\Gamma(n-k+1)}{\Gamma(n+2)}
$$
$$
= \frac{n!}{(k-1)!(n-k)!}\frac{k!(n-k!)}{(n+1)!}=\frac{k}{n+1}.
$$
$$
\bE[Y_k^2]=\frac{n!}{(k-1)!(n-k)!}\int_0^1 y^{k+1}(1-y)^{n-k} dy= \frac{n!}{(k-1)!(n-k)!}\frac{\Gamma(k+2)\Gamma(n-k+1)}{\Gamma(n+3)}
$$
$$
=\frac{n!}{(k-1)!(n-k)!}\frac{(k+1)!(n-k)!}{(n+2)!}=\frac{k(k+1)}{(n+1)(n+2)}.
$$
$\DeclareMathOperator{\Var}{Var}$. Hence
$$
\Var[Y_k]= \frac{k(k+1)}{(n+1)(n+2)}-\left(\frac{k}{n+1}\right)^2=\frac{k}{n+1}\left(\frac{k+1}{n+2}-\frac{k}{n+1}\right)=\frac{k(n+1-k)}{(n+1)^2(n+2)}.
$$
We deduce
$$
\Delta_n=\sum_{k=1}^n \Var[Y_k]=\frac{1}{(n+1)^2(n+2)}\sum_{k=1}^n k(n+1-k).
$$
We have
$$
\sum_{k=1}^nk(n+1-k)= (n+1)\sum_{k=1}^n kn-\sum_{k=1}^n k(k-1)
$$
$$
= \frac{n^2(n+1)}{2}-\sum_{k=1}^nk(k-1).
$$
Now we write
$$
k(k-1)=\frac{1}{3}\Big(\; k^3-(k-1)^3-1\;\Big)
$$
so
$$
\sum_{k=1}^n k(k-1)=\frac{1}{3}\Big(\; n^3 -n\;\Big)=\frac{n(n+1)(n-1)}{3}.
$$
We deduce
$$
\Delta_n= \frac{1}{(n+1)^2(n+2)}\left(\;\frac{n^2(n+1)}{2}-\frac{n(n+1)(n-1)}{3}\;\right)=\frac{n(n+1)(n+2)}{6(n+1)^2(n+2)}=\frac{n}{6(n+1)}.
$$
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