Suppose $n$ samples are generated from an unknown distributions and the empirical percentile is estimated.   In python:  import numpy as np n_samples = 30 desired_percentile = 90 x = np.random.uniform(0, 1, n_samples) # data from an unknown distribution perc = np.percentile(x, desired_percentile)   How can I estimate how close my estimated percentile is to the real percentile of the distribution?   I would like to estimate something like $p=P(|perc- real~perc| &gt; \delta)$ that in this example is equivalent to  $P(|perc-0.9|&gt;\delta)$.  Note that in general the $real~perc$ value is not known and that I expect $p \rightarrow 0 ~if ~ n \rightarrow\infty$  UPDATE 1  All the answers at the moment seems to suggest to estimate the confidence interval using bootstrap. Unfortunately bootstrap will work only if enough samples are available.    Suppose using bootstrap we estimate $l$ and $u$ such that:  $P(l &lt; real~perc &lt; p) &gt; 0.9$   this will not take into account the fact that the data is not enough to correctly approximate the original distribution.    Indeed for example in the case of a $n=30$ and $x_i\sim U(0, 1)$ we have:   $P(x_1&lt;0.95,\dots x_n&lt;0.95)=0.21$   as a consequence the $P(l &lt; real~perc &lt; p)$ should be less than $0.79$.  I think an estimate of the CI needs to explicitly take the number of samples $n$ as parameter.
I have a bunch of raw data values that are dollar amounts and I want to find a confidence interval for a percentile of that data.  Is there a formula for such a confidence interval?
How do high traffic sites service more than 65535 TCP connections? Show the number of up-votes and down-votes on the mobile site theme It seems we can't view the number of up-votes and down-votes a question has from the mobile site. Can this be added or is there a reason it isn't available? I was reading an example about describing homomorphisms and I'm having a bit of trouble with this one.  Show there is no group homomorphism $f : \mathbb{Z}_{10} \to \mathbb{Z}_{25}$ such that $f(1) = 3$. The first step of the proof is  $|1| = 10$ in $\mathbb{Z}_{10}$, so $|f(1)| \in\{ 1, 2, 5, 10\}$.  Does this mean that $|f(1)|$ can be $1$, $2$, $5$ or $10$? I see that $|1| = 10$ since $1$ is a generator of $\mathbb{Z}_{10}$, and that $|f(1)|$ seems to be a factor of $10$. If $f(1)$ is the image of $1$ under $f$, how would I find the order of $f(1)$?  Thanks for your help. Prove or show a counter example for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$ I'm working on my Discrete Mathematics homework and they are asking me this:  Prove or show a counterexample for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$  I'm not completely sure how to prove it. I was thinking to find a counterexample like zero but I'm not sure if that it's going to work. Or maybe try to approach it backwards?   I'm a little bit lost I would like to transfer files between two remote hosts using on local shell, but it seems rsync doesn't support synchronisation if two remotes are specified as follow:  $ rsync -vuar host1:/var/www host2:/var/www The source and destination cannot both be remote.   What other workarounds/commands I could use to achieve similar results? Is there a way to check that the conditions on your plane ticket are actually what your travel agent said they are? Proving a direct product of groups is a group I am trying to prove the following and am looking toward the math.stackexchange community to comment on whether I am on the right track or not. Thank you in advance.    Let $n \geq 1$ be an integer. Suppose $\{G_1,...,G_n\}$ is a finite set of groups. Define the set $G = G_1 \times G_2 \times ... \times G_n$ of ordered n-tuples $(g_1,g_2,...,g_n)$ where $g_i \in G_i$. Show that the operation $\bullet$: $G \times G \rightarrow G$ given by $(g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...,g'_n) = (g_1g'_1,g_2g'_2,...,g_ng'_n)$ makes $G$ into a group.    Proof of associativity: $((g_1,g_2,...,g_n) \bullet (g'_1,g'_2,...g'_n)) \bullet (g''_1,g''_2,...,g''_n)$ = $(g_1g'_1,g_2g'_2,...,g_ng'_n) \bullet (g''_1,g''_2,...,g''_n) $ = $(g_1g'_1g''_1,g_2g'_2g''_2,...,g_ng'_ng''_n)$ = $(g_1,g_2,...,g_n) \bullet (g'_1g''_1,g'_2g''_2,...,g'_ng''_n) = (g_1,g_2,...,g_n) \bullet ((g'_1,g'_2,...,g'_n) \bullet (g''_1,g''_2,...,g''_n))$  \  Proof of the existence of an identity For each $G_i \in G$, take its identity element $e_i$ and multiply it as follows: $(g_1,g_2,...,g_n) \bullet (e_1,e_2,...,e_3)$ = $(g_1,g_2,...,g_n)$.  \  Proof of the existence of an inverse: Since each $G_i \in G$ is a group, simply takes its multiplicative inverse $g_i^{-1}$ and multiply it: $(g_1,g_2,...,g_n) \bullet (g_1^{-1},g_2^{-1},...,g_n^{-1}) = (e_1,e_2,...,e_n)$. Detect whether Excel workbook is already open What is the promotion rate of professors in United States? I'm writing an essay, and want to know the promotion rate and average duration from assistant professor to associate professor and associate professor to professor in recent years. Sadly I don't know the data for the US. Could anyone tell me? One might expect the sum of the tracert latencies to equal 267ms: Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for  $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: $\ln(a+bi)=\ln[\sqrt{a^2+b^2}]+i \cdot \arctan(\frac{b}{a})$  ).  Anyway, how would I go about calculating, say, $i^j, k^i$ etc., or, more generally, $(a_1+b_1i+c_1j+d_1k)^{a_2+b_2i+c_2j+d_2k}$  (I know that exponentiating a complex number (to another non-real complex number) produces a non-unique result, so I assume the same would apply further up the hypercomplex ladder; if that's the case, I'm only concerned with the 'principal' value)?  I obviously don't want a general formula or anything like that; just some intuition and a method by which I could calculate such a thing.  And, finally (because I really like to push my luck), can this method for quaternions be extended to higher number systems (i.e. $\mathbb{O, S},$ etc.) to give an analogous result?   Thanks Book recommendation on the history of PDE/ODE? Usage and origin of "sister" in expressions like "sister company, sister ship, sister site" etc Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits. Is it possible to "hack" a MacBook so as to be able to install macOS Mojave? How to make a GIS inventory? My office will be seeing a big change in its GIS section. This section has been operational since the 1980's and has a huge collection of GIS data (i.e., shapefiles, raster files, data, etc) but never been through any inventory. Now it will happen.   Is there any automated way to extract all the information about the GIS data (i.e., shapefile, arc-info coverage, layer file, *.mxd, gdb, raster file, and more) from a PC to an Excel file? The information may include date of creation, date last edited, folder or container name, etc. Fisher's exact test on 2x4 table I am working on 4 different plants. I have the (RNA-Seq) data from sequencing. I look for two events E1 and E2, say, at certain positions in their genome. Let's say E2 is the common one and E1 is a special event. The positions I look for are identical across all 4 (in the genome). And let's say I see that the observation I have for 1 particular position is:      P1   P2  P3  P4 E1   0   20   0   17 E2 100   80 100  120    Here, P1 through P4 refers to plants and E1 and E2 refers to the events. E2 is more common. So, my objective is to actually check if E1 occurs more often in one or more plants than in the others. If they occur at the same proportion in all, of course it is not interesting to me.  I have 2 questions: (I have already asked  before but didn't get an answer)     Will a fisher test be right for this problem?    I hypothesize (Null) that the proportion of E1 is not different in occurrence between the 4 plants. Now, I set out to find if the proportion I have here is by any means significantly different. I use R, fisher.test() and I get p-value=2.5e-10. So, I reject my Null hypothesis for this case because I find strong evidence against it.  Sometimes, I have an observation like this,        P1   P2  P3  P4 E1   0   20   0  17 E2   0   80   0 120    then fisher.test() gives me a p-value of 0.147. So, I don't reject the Null   hypothesis. However, from a biological point of view, I would consider this significant. However fisher test answers the question I originally asked. I guess the proportion 0/0 for P1 and P3 are not useful (or not used).  So, my question is: Is it possible to modify the test such that it is sensitive even if E1 and E2 are 0 in 1 or more plants for a particular observation?     Having thought a bit to frame this post, I guess, in that case I have to ask a different question. Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic. Is there a better (or other) way(s) to prove the following statement? Also, the same argument works for multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{Q}-\{0\}$, right?       Problem Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.   Solution By cantor's diagonal argument, there is no possible bijection between $\mathbb{Q}$ and $\mathbb{R}$. Since an isomorphism needs to be a bijection, there is no possible isomorphism between the additive groups $\mathbb{R}$ and $\mathbb{Q}$.    Thanks This is question 2.9 #9 from Topics in Algebra by Herstein:     If $o(G)$ is $pq$ where $p$ and $q$ are distinct prime numbers and if $G$ has a normal subgroup of order $p$ and a normal subgroup of order $q$, prove that $G$ is cyclic.   Let $N$ denote the normal subgroup of order $p$ and $M$ denote the normal subgroup of order $q$. Here are a couple things I noted while exploring the problem:   $N$ and $M$ are cyclic, since they have prime order. $G = NM$ [1].   Any hints where to look next?    [1] Since $N,M$ are normal, $NM \le G$. Then $o(NM) \mid o(G)$, so $o(NM)$ is either $1$, $p$, $q$, or $pq$. The first three don't work because they force one of $N$ or $M$ to be $(e)$, so we have $o(NM) = pq$. Therefore, $NM = G$. Kids' fantasy movie from the 80s or 90s?