Questions tagged [median]
For questions about the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.
388
questions
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Is the median of a discrete distribution with an infinite population a computable number?
For that matter, I ask for any quantile.
I know that taking larger and larger sample sizes can get me closer to the true median, but does this satisfy the definition of Computable number?
0
votes
1answer
25 views
General Expectation Property for Gaussian
It is a well known property that $E[|X - p|]$ is minimized for $p$ such that $P(X \leq p) = \frac{1}{2}$ (i.e. $p$ is the median value for the random variable $X$).
Now suppose $\mathbf{X} \sim \...
1
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2answers
35 views
Which of the statements are correct
I was trying to answer the question below.
Given the sets S1 = {2,3,5,8,9} and S2 = {3,4,5,6,7} which of the following statements are correct?
a. Median value and mean are 5 for both sets
b. For S2 ...
0
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0answers
23 views
Median of a difference of random variables.
Let $X,Y$ two real random variables and let $Med(X)$ the median of $X$ and $Med(Y)$ the median of $Y$. We have that $\mathbb{P}(X\geq Med(X))\geq 1/2$ and $\mathbb{P}(Y\geq Med(Y))\geq 1/2$. My ...
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23 views
Convergence almost surely implies convergence of the medians.
Let $\{X_{n}\}_{n}$ be a sequence of real random variables and let $X$ be a random variable such that $\lim_{n}X_{n}(\omega) = X(\omega)$ for all $\omega\in\Omega$. It implies that $X_{n}$ converges ...
0
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1answer
16 views
Convergence of Random Variables and Median.
Let $\{X_{n}\}_{n}$ be a sequence of real random variables and let $X$ be a real random variable such that $\lim_{n} X_{n}(\omega) = X(\omega)$. Assume that $X_{n}(\omega)$ is a monotone decreasing ...
0
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0answers
15 views
Range of the median for a $Bin(n,p)$ distribution
I came across a question which went like this:
Show that the median($m$) of any $Bin(n,p)$ distribution is such that $[np]\leq m \leq [np]+1$.
I tried to prove this by the very basic notion of a ...
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0answers
14 views
Mean/median/etc of a zero-mean random variable with exponent
Suppose $\epsilon \sim N(0,\sigma^2)$, $a\in(0,1)$ and $x>0$ are constant. Is there any way of estimaing the following:
$$(\frac{x+\epsilon}{x})^a$$
Although $\mathbb{E}[\frac{x+\epsilon}{x}]=1$, I ...
1
vote
1answer
50 views
Probability - median and mean
I'm having a difficult time with this one. Would be happy for some help.
I'm kind of gets why its true but can't manage to prove it.
Let $X$ be a random variable with median $\Bbb MX$ such that ...
0
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1answer
20 views
weighted median, but manually typed weights, not frequencies
Since theres some contoversy about the definition of the weighted median, I wonder if my doing is even possible:
I have a large 2d matrix
...
2
votes
0answers
34 views
How to compute median without storing all the values?
I have a source of data that continuously gives me measurement results. At some point I compute the mean. Luckily to this end, I don't need to store all the values, I only store the sum and the number ...
0
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0answers
9 views
Change on individual medians affect which of the following measures
The median of 21 distinct median was increased by 1, which of the following measures is not effected
by this change?
(A)Mean. (B)Mean deviation about median. (C)Mid-range. (D)Standard deviation.
The ...
3
votes
1answer
22 views
Infimum of the set of medians.
Let $\{F_{n}\}_{n}$ be a sequence of cumulative distribution functions such that converge to $F$, in the sense that $F_{n}(x)\rightarrow F(x)$ for all $x\in\mathbb{R}$. We define the function infimum ...
0
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1answer
39 views
Confidence Interval for the Median of Any Continuous Distribution
This is an example 4.4.7 from "Introduction to Mathematical Statistics" by Hogg, McKean, and Craig.
Let $X$ be a random variable of the continuous type with cdf $F_{X}(x)$ and let $\xi_{1/2}$...
0
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1answer
25 views
Calculating the median from a sample
Say I have a sequence $a_1\leq a_2\leq\dots\leq a_n$ of $n$ numbers. Say I pick a subsequence of $k$ samples from this sequence. Can I approximate the median of the original sequence from the sample?
...
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0answers
12 views
How to calculate the sample variance if the data set is changed
I have a $15$ samples data-set in which
$$\bar x= 15, s^2 = 3.2, š_0 = 11, š_1 = 14.0, š_2 = š_4 = 17$$
However, the exact number of each element is unknown.
Now, I have added two new sample in ...
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0answers
15 views
Generating log normal distribution using median and 95th percentile
I have median and 95th percentile value of certain data which might follow log normal distribution, but I don't have that "data" ,I only have these two aforementioned aspects of data. The ...
0
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0answers
31 views
Limit of the Median of Random Variables
Let $X_{1},X_{2},\cdots$ be real random variables identically distributed. We consider the sequence $m_{n} := Med(X_{n})$, where $Med(\cdot)$ denotes the Median of a random variable. My question is ...
1
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2answers
45 views
Solve for the base of an isosceles triangle
The area of an isosceles triangle is $S$ and the angle between the medians to the legs, facing the base, is $\alpha$. Find the base of the triangle.
Let $CH$ be the third median of the triangle. The ...
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0answers
31 views
Median instead of mean distance of an underdetermined system of linear equations
I am given a number $N$ of 2D point pairs $(x_i,y_i)$ with corresponding values $(v_i,w_i)$. My task is now to find a linear matrix s.t.
$$ \begin{pmatrix} m_{x,1} & m_{y,1} & b_1 \\ m_{x,2} &...
3
votes
4answers
93 views
Find the ratio $\frac{AF}{FC}$
In the figure below, $AD$ is the median on $BC$. The point $E$ divides $A$ and $D$ in the ratio $1:2$. $BE$ produced meets $AC$ at $F$. Find the value of $AF:FC$
My try: I joined $E,C$.Let area of $\...
0
votes
1answer
17 views
Do CEVIANS in Triangles divides the emerging angle in the same ratio as they do to the opposite side .
Ok I know that suppose in a triangle ABC , there's a cevian AD intersecting BC at D and dividing it in the ratio 4:5 so which implies that it divides Area of Triangle also in ratio 4:5 by that cevian. ...
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vote
1answer
71 views
Show that $|\operatorname{median}(X) - \operatorname{mean}(x)| \leq \sigma_X$ [duplicate]
Let $m$ denote median and $\bar{x}$ denote mean and $\sigma$ denote the standard deviation, I want to show that $|m - \bar{x}| \leq \sigma$. Since the LHS and RHS are both positive, we can prove $(m - ...
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0answers
20 views
Mean vs Median for Customer Valuation
This question is just as much of a math one as it is a valuation/business one, but here it goes:
Say I have a heavily right skewed distribution (cumulative order amounts from customer over their 'life'...
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0answers
16 views
What seems more useful for this particular problem, Mean or Median and Why?
You are given an array of positive integers. You are told to make all the numbers equal by doing this operation, i.e. to increase/decrease the value of an array element. The cost of the operation will ...
1
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1answer
69 views
Find the area of a triangle $\triangle FGH$ [closed]
In triangle $\triangle FGH$, $GM$ is a median that lies on $4x-y=27$; height $HA$ lies on $x-y+3=0$; $F$ is $(4,5)$. Find the area of the triangle $\triangle FGH$.
My attempt:
F is not on any of the ...
0
votes
3answers
59 views
What is mean and median of $X$ if $P(X \geq \frac{1}{2}+x)=P(X \leq \frac{1}{2}-x)$?
Let $X$ be a random variable such that $E[X] < \infty$ and $P(X \geq \frac{1}{2}+x)=P(X \leq \frac{1}{2}-x), \forall x \in R$ then find $E(X)$ and median$(X)$.
My attempt:
$\int_{-\infty}^{\frac{...
3
votes
1answer
74 views
Variance of a sample median
Suppose $X_1,\ldots,X_n\sim\text{i.i.d.}\operatorname N(\mu,\sigma^2).$
I think I've only ever seen one way to prove that the sample mean of $X_1,\ldots,X_n$ has a smaller variance than does the ...
3
votes
1answer
107 views
median bisector?
The point $O$ is the incenter of the triangle $ABC$, $N$ is the centroid. Then it is possible to construct the point $M$ that can be loosely defined as the intersection point of the "median ...
0
votes
1answer
66 views
Bayes estimate under absolute loss function [closed]
Consider a Bayesian model where the prior distribution is $\lambda(\theta) = \frac{2}{(1+\theta)^3}$, $\theta > 0$ and $X\sim U(0,\theta).$ Find the Bayes estimate $\delta(x)$ if the loss function ...
1
vote
1answer
28 views
k-mean clustering minimize L1 distance
k-mean clustering minimize L1 norm
In k-mean clustering, if I want to minimize the L1 distance from any point to cluster center, the error function and derivative is shown above.
However, according to ...
0
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1answer
41 views
Extension of median minimizing the sum of absolute deviations (the $L_1$ norm)
This is an extension of the question asked in The Median Minimizes the Sum of Absolute Deviations (The $ {L}_{1} $ Norm)
. Except with the extra constraint that $x \in S$.
The solutions provided there ...
3
votes
1answer
50 views
Algorithm to compute “median”
Let $x_1,\dots,x_m\in\mathbb{R}^d$ be a finite set of points. I define the $d$-dimensional "median" $y\in\mathbb{R}^d$ to be the point minimizing the sum of distances to $x_j$,
$$
y = \arg\...
0
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1answer
25 views
Median of a continuous random variable
Consider the cdf $F(x)=1-e^{-x}-xe^{-x}, 0\leq x <\infty$, zero elsewhere. Find the median of this distribution,
The given CDF is a complicated and I am finding it difficult to find x for which P(X&...
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0answers
37 views
median of a discrete random variable
Find the median of the following distribution
$p(x)=\frac{4!}{x!(4-x)!} {(\frac{1}{4})}^x {(\frac{3}{4})}^{4-x}$ , x=0,1,2,3,4, zero elsewhere.
In the question, the median is defined as the value of x ...
0
votes
1answer
34 views
The method of moments estimator of median
In some textbook, it is written that the method of moment estimator can be applied even in estimating median. However, I can't come up with the idea.
In many cases the median is estimated by $x_{n/2}$...
2
votes
2answers
97 views
Triangles and medians
Let G be the ABC barycenter. A line intersects the medians AD, BE and CF in X, Y and Z, respectively. Prove that
$$\frac{XD}{XG}+\frac{YE}{YG}+\frac{ZF}{ZG}=3$$
By areas relations I found that
$\frac{...
0
votes
2answers
26 views
Median (and consequently the mean) of an evenly-spaced list [closed]
Why is it the case you can find the median (and consequently the mean) of an evenly-spaced list by taking the mean of opposite terms? Where opposite refers to opposite positions (e.g. first and last). ...
0
votes
1answer
45 views
Find the density of the median of $2n$ i.i.d $U([0,1])$ random variables
As a part of a probability problem I found the PDF and CDF of the $i^{th}$ order statistic in a sample.
When told that $$X_1,...,X_n{\sim}^{i.i.d}F$$ where F is countinuous, so I got to the conclusion ...
1
vote
1answer
38 views
Convention of finding Quartiles
Find the quartile deviation for the data
$$
\begin{array}{|c|c|c|c|}
\hline
x& 2 & 3 & 4&5&6 \\ \hline
f& 3 & 4 & 8&4&1\\ \hline\end{array}
$$
My Attempt
$$
\...
1
vote
1answer
115 views
Area of a triangle as a function of its bisectors.
Area of a triangle can be easily calculated using only its 3 medians
is it also possible to find its Area as a function of its 3 bisectors?
Lots of people tried to find the solution and oddly ...
0
votes
1answer
26 views
Find median of a combination of tables.
Values in List $P$ are distinct from values in List $Q$. List $R$ consists of all the values in list $P$ and all the values in list $Q$.
Find the median of list $R$.
My logic:
Here the total number ...
0
votes
1answer
69 views
Mean deviation gives better results when deviations are taken from the median, why?
The mean deviation gives better resuls when deviations are taken from the median instead from the mean, because the sum of the deviations from the median is less than the sum of the deviations from ...
0
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1answer
37 views
Find global median from medians of subgroups
Supposte I have a list of numbers $\{x_1, \ldots, x_N\}$, not necessarily ordered, and I divide it in subsets $\{ \{x_1,\ldots,x_{d_1}\}, \{x_{d_1+1},\ldots,x_{d_2}\}, \ldots \}$, where $d_n$ is the ...
0
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0answers
27 views
How to derive formula of variance of sample median?
I wonder how to derive the formula of variance of median of sample.
I searched it for a day but couldn't find, so I'm asking help here. You may just notice me the link of it.
I saw the result form is ...
4
votes
0answers
47 views
The medians in a right triangle
$\triangle ABC$ is a right triangle $(\measuredangle ACB=90^\circ)$. The medians are $AA_1=4\sqrt{13}$ and $BB_1=2\sqrt{73}.$ Find the sides of $\triangle ABC$.
By the Pythagorean theorem: $\triangle ...
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1answer
22 views
How to find the probability of the number of data
I have an exercise, i try to finish all of the question. I stack for the last one. Anyone can help me?
I attach the question.
I find the solution for (i) 2
(ii) 2.63 (iii) 0.055
And i need your help ...
0
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1answer
53 views
Median estimator; Theoretical statistics, Keener
I am reading the book Theoretical Statistics: Topics for a Core Course from Keener, 2010. In section 8.4 Medians and Percentiles includes:
"Let $X_{1},...,X_{n}$ be random variables. These variables, ...
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0answers
20 views
Fitting lognormal distribution from D10, D50, and D90
I have a statistical sample where D10 = 8 (10% of population under the value 8), D50 = 11 (median), and D90 = 18 (90% of population under de value 18)
Now, I need to fit the $\mu$ and the $\sigma^2$ ...
0
votes
1answer
31 views
Deriving median from minimum and range
Let's say you have an odd number of n distinct integer and you know the range of these integers. (E.g. your min integer is 10 and your max integer is 50, this implies that the range is 30)
Is there a ...
