How to prove the equation below in a simple way? $$\min_{\mu}\sum_{i=1}^n|y_i-\mu|=\text{median}\{y_1,\cdots,y_n\}$$
49109
Suppose we have a set $S$ of real numbers. Show that   $$\sum_{s\in S}|s-x| $$   is minimal if $x$ is equal to the median.    This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.
32235
Please. I am trying to understand the proof, that cdf of minimum of $n$ random variables is $1-[1-F(x)]^n$  If I have $n$ independent random variables $X_1, \dots, X_n$, all of them have the same CDF $F(x)$, what is the point of searching for the $\min=\{X_1, \dots, X_n\}$? Shouldn't $X_1, \dots, X_n$ have the same value for every x? (In the meaning: $P (\min \{X_1, \dots, X_n\}≤x) = P (\bigcup_{i=1}^n X_i\}≤x) \cdots$)  Thanks.
14131
How can I prove that the minimum of two exponential random variables is another exponential random variable, i.e. Z = min(X,Y)
30138
Can somebody explain me clear the mathematical logic that would link two statements (a) and (b) together? Let us have a set of values (some distribution). Now,  a) Median does not depend on every value [it just depends on one or two middle values]; b) Median is the locus of minimal sum-of-absolute-deviations from it.  And likewise, and in contrast,  a) (Arithmetic) mean depends on every value; b) Mean is the locus of minimal sum-of-squared-deviations from it.  My grasp of it is intuitive so far.
30402
I need hint proving this $$\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$$
82294
I have a simple question, but it is hard to google it. I have this equation here:  $$y(t, x) = \sum_{i=1}^{d}(|x_i| \wedge t)^{2} $$  Here $x$ is a size $d$ signal and $t$ is just a scalar. I am not sure how to read that equation in english... I understand everything except for how they use the $\wedge$ here...  If context helps, this is part of a cost function, based on a threshold $t$ that is selected, for your vector $x$.
14798
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
20622
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
74663
Let $X_1,X_2,\ldots,X_n$ and $A_1,A_2,\ldots,A_n$ be real numbers taken at random, and $\alpha\gt0$. Show that: $$\sum_{k=1}^n |X_kA_k| \le \frac{1}{\alpha}\sum_{i=1}^n X_k^2 + \frac{\alpha}{4}\sum_{k=1}^n A_k^2$$
87410
Can I simplify or approximate this equation without sigma and combination?  \begin{align} \sum_{i = 0}^n (-1)^i {n \choose i} \frac{{d+1}}{d(di + 1)} \end{align}
48160
I want to solve the following problem in T.Tao's random matrix theory book. Let $X$ be a random variable with finite second momment. A median $M(X)$ of $X$ saisfies $\mathbb{P}(X&gt;M(X)),\mathbb{P}(X&lt;M(X))\leq 1/2$. Now I wan to prove that for any median: $$M(X)=\mathbb{E}(X)+O((\mathbb{Var}(X))^{1/2})$$  I think we have to use Chebyshev's inequality for this, which I have done to obtain: $$\mathbb{P}(X&gt;\lambda \sigma)\leq \mathbb{P}(|X|&gt;\lambda \sigma)\leq \frac{1}{\lambda ^2}$$ where $\sigma=(\mathbb{Var}(X))^{1/2}$. But I fail to interprete this...
116810
How to show  $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
54001
I am trying to show $$\sum_{d \leq x} \mu(d)\left\lfloor \frac{x}{d} \right\rfloor = 1 \;\;\;\; \forall \; x \in \mathbb{R}, \; x \geq 1 $$ I know that the sum over the divisors $d$ of $n$ is zero if $n \neq 1$. So we can rule out integers that are divisors of $x$ (if $x &gt; 1$). I am not sure if I am on the right track to prove this. Any hints would be helpful. This is homework.
29883
How is it possible to evaluate:  $$\sum_{k=1}^n{((-1)^{n-k}\cdot k^3)}=n^3 - (n-1)^3 + (n - 2)^3 - \cdots \pm 1^3$$  The fact that there is the $\pm$ at the end makes it difficult.
70092
I am trying to show that: \begin{equation} \frac{n}{\phi(n)} = \sum_{d \mid n} \frac{\mu^2(d)}{\phi(d)} \end{equation} where $\phi(n)$ is  and $\mu$ is the .  The identity clearly holds for $n=1$, so if we write $n=p_1^{a_1} \cdots p_k^{a_k}$ for the prime decomposition of $n$, the left-hand side becomes: \begin{eqnarray*} \frac{n}{\phi(n)} &amp; = &amp; \frac{p_1^{a_1} \cdots p_k^{a_k}}{\phi(p_1^{a_1}) \cdots \phi(p_k^{a_k})} \\ &amp; = &amp; \frac{p_1^{a_1} \cdots p_k^{a_k}}{(p_1^{a_1-1})(p_1-1) \cdots (p_k^{a_k})(p_k-1)} \\ &amp; = &amp; \frac{p_1 \cdots p_k}{(p_1-1)\cdots (p_k-1)} \end{eqnarray*}   Whereas the right-hand side becomes:  \begin{eqnarray*} \sum_{d \mid n} \frac{\mu^2(d)}{\phi(d)} &amp; = &amp; \sum_{\substack{d \mid n \\ p^2 \nmid d}} \frac{1}{\phi(d)} \\ &amp; = &amp; 1 + \frac{1}{\phi(p_1)} + \dots + \frac{1}{\phi(p_k)} + \frac{1}{\phi(p_1 p_2)} + \dots + \frac{1}{\phi(p_{k-1}p_k)} \\ &amp; + &amp; \dots + \frac{1}{\phi(p_1 \cdots p_k)} \\ &amp; = &amp; 1 + \frac{1}{p_1 - 1} + \dots + \frac{1}{p_k - 1} + \frac{1}{(p_1 - 1)(p_2 - 1)} + \dots + \frac{1}{(p_{k-1}-1)(p_k-1)} \\ &amp; + &amp; \dots + \frac{1}{(p_1 - 1) \cdots (p_k - 1)} \end{eqnarray*}  I am unsure of how to proceed from this point however. Any help would be much appreciated.
76504
I'm struggling with proving the following inequality: $$\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$$  for all $t \geq 2$. I think it is monotonic non-increasing in $t$, which would suffice.  Thanks in advance!
114521
prove the following equation : $$\sum_{1\le i\le r}\frac{i+1} { r+1}{2r-i\choose r-i}{s+i-2\choose i}+\frac{1}{r+1}{2r\choose r}={s+2r-1\choose r} - {s+2r-1\choose r-1}$$