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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.

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Median of a Poisson-Binomial distribution

Let $Z$ be a random variable following a Poisson-Binomial distribution with parameters $(p_1, \dots, p_n)$ such that $\sum_{i=1}^{n}p_i > \frac{n}{2}$. Consider the following two properties: -If $n$...
Ibra's user avatar
  • 175
3 votes
0 answers
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Number of possible histograms of a test given that the median is 60

There is the following problem I'm having a hard time coming up with a Sigma-less solution to: If $360$ have taken a test, and grades can range from $0-100$ (inclusive), how many possible histograms ...
natitati's user avatar
  • 393
1 vote
1 answer
61 views

Inequality with medians

It is well known that in any triangle $\triangle ABC$ with side lengths $a,b,c$ and medians $m_a,m_b,m_c$ inequality \begin{equation} \frac{3}{4}(a+b+c)<m_a+m_b+m_c<a+b+c \end{equation} holds. ...
Oliver Bukovianský's user avatar
1 vote
4 answers
95 views

A possible error in a math question of 2022 Brazil's ENEM, a equivalent to US's SAT

This is a rather simple question, and it may seem like a physics question, but it actually belongs to a math test. I'll translate it into English first and then reproduce the Portuguese version. "...
Igor Paulino's user avatar
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1 answer
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Median of Mixed Random Variable [closed]

I have the following CDF $$ F_X(x) = \begin{cases} 0 & x < 0 \\ 1 - p e^{-x} & x \geq 0 \end{cases} $$ I found $\mathcal{X} = \{ 0\} \hspace{0.1cm} \cup (0,\infty)$ with $P(X = 0) = 1-p$ ...
daniel's user avatar
  • 753
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2 answers
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Finding the distribution of the median of three independent random variables

Problem: Let $Y_1$, $Y_2$ and $Y_3$ be independent and uniformly distributed over the interval $(0,1)$. Let $Y_0$ be the median of the three variables. Find the probability density for $Y_0$. Answer: ...
Bob's user avatar
  • 4,064
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Medians from rows of numbers

You have three rows of numbers, $(a_1,a_2,a_3), (b_1,b_2,b_3), (c_1,c_2,c_3)$, each in the range $[0,1]$ and summing up to $1$. If the medians of the three columns are $m_1,m_2,m_3$ with sum $m$, you ...
user355066's user avatar
1 vote
1 answer
82 views

The Medians of Lipschitz Functions on $(X,d,\mu)$ (Existence and Uniqueness)

Let $\varphi:(X,d,\mu)\to \Bbb R$ be a Lipschitz function, where $\mu$ is a probability measure on the metric space $(X,d)$. The median $m_\varphi$ of $\varphi$ is defined as the real number such that ...
stoic-santiago's user avatar
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median of multivariate Gaussian

I have the following basic question. I am interested in the spatial median of mutlivariate Gaussian random variable. Let $d$ be the dimension and let $\mathcal{N}(\mu,I_d)$ denote the multivariate ...
MMH's user avatar
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Finding the median value, am I missing something?

If I'm not missing anything, please tell me. This is my understanding of the definition of a statistical mean: Using the definition: The median is the middle value that separates the lower and higher ...
Terry Wendt's user avatar
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1 answer
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Maximum Likelihood Estimation of median for an exponential distribution

Given data x1, ... xn i.i.d. with exponential distribution and unknown parameter λ, determine maximum likelihood estimation of θ given the observed data where theta is the median of the distribution. ...
Michael Williams's user avatar
2 votes
2 answers
79 views

What is the probability that a subset of size $m$ will have the same median as the set of size $n$?

I feel like I have been bashing my head against the wall on something that I thought would be easy. Not from a math background so this might actually be trivial. I have a set of numbers $S$ with size $...
Gad Raganas's user avatar
1 vote
1 answer
56 views

Why I am getting two different medians

let's say we have: $10, 12, 15, 10, 10, 12$ sorted: $10,10,10,12,12,15$ this way is clear that the median is $\frac{10+12}{2} = 11$ however, I was told that I could find the median using this table ...
samsamradas's user avatar
3 votes
0 answers
81 views

Is the median a measurable function of the probability distribution?

For $\mu \in \mathcal P(\mathbb R)$, let $m(\mu)$ be the median of $\mu$, defined as the smallest of all medians of $\mu$ as follows: $$ m(\mu) = \inf \left\{ x \in \mathbb R \,\middle|\, \mu((-\infty,...
Cyril B.'s user avatar
  • 115
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Distance of a hyperplane from Geometric Median

Let $d,n \in \mathbb{N}$. Assume we have $n$ points, $x_1,\dots,x_n$ where for every $i \in \{1,\dots,n\}$, we have $x_i \in \mathbb{R}^d$. Define the geometric median as $$ \theta^\star \in \arg\min_{...
MMH's user avatar
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1 answer
120 views

Find recurrence relation for sum of absolute deviations in a sequence

Given an ordered sequence $S_n = s_1, s_2, ..., s_n$ we define its cost as the sum of absolute deviations from any median: $$ cost(S_n) = \sum_{i=1}^{n} \left| s_i - median(S_n) \right|\text{, where } ...
Andrei Onoie's user avatar
2 votes
0 answers
110 views

Exact bounds for the Median of the Hypergeometric distribution

I am wondering if anybody knows good exact bounds for the Median of a Hypergeometric distribution: $$ X \sim \text{Hypergeometric}(N,K,n) $$ One exact bound could be given by exploiting the inequality ...
Konczer J's user avatar
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199 views

Which class will be the median class if n/2 is the cumulative frequency of a class interval?

An example problem: When using the textbook formula, Median = L + {[(n/2)-c.f.]/f}*h, where L is the lower limit of the median class, n is the total number of observations, c.f. is the cumulative ...
Curious 9th Grader's user avatar
3 votes
0 answers
55 views

What is the intuition behind medians (and quartiles) being defined as the value of the $(n+1)/p$ th observation?

When calculating the median of an un-grouped distribution, we take the $(n+1)/2$ the value. For quartiles it's $\text{I} \cdot (n+1)/4$. For deciles it's $\text{I} \cdot (n+1)/10$. What is the ...
SeetheMoar's user avatar
4 votes
2 answers
70 views

The rate at which the expectation of the square of the empirical median of i.i.d. $[-1,1]$-valued uniform random variables goes to zero

Suppose that $X_1,X_2,\dots$ is an i.i.d. sequence of $[-1,1]$-valued uniform random variables. Let $\bar{X}_t$ be the empircal mean of the sample $X_1,\dots,X_t$. Then $\mathbb{E}\big[|\bar{X}_t|^2\...
Bob's user avatar
  • 5,783
1 vote
1 answer
72 views

Finding median from PDF and multiple cases

The random variable $X$ has the probability density function given by: $$ f(x) = \begin{cases} \frac{2x}{3}, & 0 \le x \le 1 \\ \frac{2}{3}, & 1 \lt x \le 2 \\ 0, & \text{otherwise} \...
hohner's user avatar
  • 1,049
3 votes
0 answers
50 views

How to simplify "median of medians" formula?

Given a nested median formula: $ \operatorname{med}\big( \operatorname{med}(a_{1,1},\ldots,a_{1,n}), \operatorname{med}(a_{2,1},\ldots,a_{2,n}), \ldots, \operatorname{med}(a_{k,1},\ldots,a_{k,n}) \big)...
Erel Segal-Halevi's user avatar
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0 answers
29 views

Robustness of Geometric Median to Change of k Points

Let $N \in \mathbb{N}$ be a constant. Let $B_d(1)$ denote the ball of radius one in $\mathbb{R}^d$. Let $(a_1,\dots,a_N)\in (B_d(1))^N$ and $(b_1,\dots,b_N)\in (B_d(1))^N$ be two sets of N data ...
MMH's user avatar
  • 714
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59 views

Characterize a function given the average and the median

How to (1) obtain some examples for an $f$ function and/or (2) characterize the whole set of $f$ functions which has the following properties given $m$ ($\leftarrow$ the median) that $0 < m < ...
Bálint Sass's user avatar
2 votes
1 answer
50 views

Deducing an incorrect statement when the mean and median are given

$~a,~b,~c,~d$ are positive integers such that $a~<~b~<~c~<~d$ If mean and median of $~a, ~b, ~c,~d$ are $35$ and $39$ respectively, then which one of the following statements cannot be true? ...
Darshit Sharma's user avatar
8 votes
1 answer
203 views

How to approximate the median of the numbers in the first $n$ rows of Pascal's triangle?

How can we approximate the median of the numbers in the first $n$ rows of Pascal's triangle? (The top row is the $0$th row.) Using Excel, I made a graph of the natural log of the median against $n$, ...
Dan's user avatar
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0 votes
1 answer
34 views

Finding value of m, where m is the bound of a pdf such that it gives the median value.

I have come across this question whilst doing my homework and would like to clarify on the value of m. The median of a continous random variable is given below. $\int_{-\infty}^{m}f(x)dx = \frac{1}{2}$...
Catt's user avatar
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2 votes
0 answers
204 views

distance between coordinate-wise median and geometric median

Let $N \in \mathbb{N}$ be a constant. Let $B_d(1)$ denote the ball of radius one in $\mathbb{R}^d$. For $i \in \{1,\dots,N\}$, let $a^{(i)}$ be a vector in $B_d(1)$. Define the geometric median of $a^...
MMH's user avatar
  • 714
7 votes
2 answers
268 views

If $a_1=1, a_n=|\cot a_{n-1}|$, then what is $\lim\limits_{n\to\infty}\text{median}\{a_1,a_2,\dots,a_n\}$?

I made up the following sequence: $a_1=1, a_n=|\cot a_{n-1}|$. What is $\lim\limits_{n\to\infty}\text{median}\{a_1,a_2,\dots,a_n\}$ ? My thoughts Here is the graph of $a_n$ against $n$, for $1\le n \...
Dan's user avatar
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0 answers
26 views

What is the term for median but use the geometric mean instead of arithmetic mean if there are an even number of values?

"Geometric median" seems like the obvious choice for "median (but for even number of values, take the geometric mean of the central two values instead of the arithmetic mean)", but ...
DanielM's user avatar
3 votes
1 answer
117 views

Is it possible to determine sides in a triangle given angle of median and opposite side length?

Questions In the triangle $ABC$ below we know the values of $\alpha$, $\beta$ and $\vert BD\vert$ in addition to knowing that AD is the median in the triangle, i.e. $\vert BD\vert = \vert DC\vert$. Is ...
GraftCraft's user avatar
4 votes
2 answers
592 views

Mean of Medians? Median of Means?

This is a question I have wondered about for a long time and have never been able to find a full mathematical explanation behind this. Suppose there are 100 countries. As an experiment: We give ...
stats_noob's user avatar
  • 3,250
0 votes
0 answers
34 views

Finding the median

I have a following problem to solve: Let X be a random variable equal to a greater dice number when throwing two symmetric dices with values from 1 to 6. What is the median of X? I'm not quite sure if ...
Karol Wyrębkiewicz's user avatar
0 votes
0 answers
39 views

Can we find the probability using the mean and median ? If it is possible, how can we calculate it?

Lucas is a student who did a math competition against 2875 other people. His rank in his results will allow him to have a school. Everybody has distinct ranks. The nearest of 1 is better. The result ...
Jotadiolyne Dicci's user avatar
0 votes
0 answers
44 views

mean median and maximum product question

The mean and the median of nine distinct positive integers is 30. If the nine integers are such that their product is maximum, what is the product of the smallest and the largest integers? My attempt: ...
CountDOOKU's user avatar
  • 1,051
0 votes
0 answers
13 views

Statistical indexes order in an asymmetrical distribution

I was wondering what's the order between mode, median, arithmetic mean, geometric mean and harmonic mean in a distribution with positive skewness and in another with negative skewness. Which are the ...
Davide Barcella's user avatar
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0 answers
35 views

Median definition

Definition 6.1.1 (Median): We say that $c$ is a median of a random variable $X$ if $\mathbb{P}(X≥m)≥ 1/2$ and $\mathbb{P}(X≤m)≥ 1/2$. (The simplest way this can happen is if the CDF of $X$ hits $1=2$ ...
Saif's user avatar
  • 1
1 vote
1 answer
93 views

Consider equilateral triangle ABC and points D and E such that $BD = \frac{BC} 3$. If DF is perepndicular on AD, F belongs to AC, prove that EF = AD.

Consider equilateral triangle ABC and points D and E such that $BD = \frac{BC} 3$, DE belongs to (BC), and E is the midpoint of (AD). If DF is perepndicular on AD, F belongs to AC, prove that EF = AD. ...
user avatar
0 votes
0 answers
33 views

Looking for a distribution that mean and median are different.

I am looking for a distribution whose mean and median are different, preferably a distribution that share some properties with Gaussian distribution. I kinda know that there is something called skewed ...
Mq Hu's user avatar
  • 56
0 votes
1 answer
222 views

Expectation of median of independent and identically distributed exponential random variables

Suppose there are $n=2k+1$ independent and identically distributed exponential random variables with rate parameter $\lambda.$ Find $\mathbb{E}[median(X_1,\cdots,X_n)]?$ To do so, I found the density ...
sunspots's user avatar
  • 802
0 votes
1 answer
116 views

Probability of the median of three independent outcomes

I have come across a statistics problem that I want to solve but I don't know where to begin. The problem is the following What is the probability that the median of three random numbers between 60 ...
Keudn's user avatar
  • 11
0 votes
1 answer
50 views

If the median of $X$ is greater than the median of $Y$ then $\Bbb P(X>Y) \geq \frac14$

The median value of a random variable $X$ is $m$ if $\Bbb P (X \geq m) \geq \frac12$ and $\Bbb P (X \leq m) \geq \frac12$. Let $X$ and $Y$ be independent real random variables. Show that if the median ...
Ava Kate Lich's user avatar
1 vote
4 answers
273 views

HELP Let ABC an isosceles right triangle with ABC = 90°. Consider the point P on AB such that $\frac{PA}{PB}=2$. Show that $PA\cdot BP=BQ\cdot BG$.

Let ABC be an isosceles right triangle with ABC = 90°, M and N the midpoints of sides BC and AC, and G is BN intersected by AM. Consider the point P on AB such that $\frac{BM}{BP}=2$. If BN intersects ...
user avatar
1 vote
0 answers
61 views

Maximum Likelihood Estimation (absolute value)

Im currently struggling on a problem. Let X1, . . . , Xn be a random sample of a random variable that has a pdf: $$f(x)=\frac{1}{2}\cdot e^{-\left |x-\theta \right |}, \infty <x<\infty $$ And ...
Jacob Larsson's user avatar
1 vote
2 answers
102 views

HELPP... Prove the equivalence: ACB = 30 only if BD = 2*MD

Please help mee!!!!!! In triangle $ABC$, $BAC=90^\circ$, $M$ is its middle $ [BC]$ and $BD\perp AM$, $D\in(AC)$. Prove the equivalence: :: $$ ACB = 30^\circ \iff BD = 2\cdot MD $$ You can see my ...
user avatar
0 votes
1 answer
122 views

Finding the median of the total number of die rolls

Question A fair die is to be rolled repeatedly until a six comes up. Find the median of the total number of rolls given that five comes up on the first roll. My working Clearly, this follows a ...
Ethan Mark's user avatar
  • 2,187
2 votes
0 answers
310 views

Median of three numbers... but not the typical way

Let's say I have three numbers $a$, $b$, and $c$. To find the median of these three numbers, you order them from least to greatest, then take the second (middle) number. For example, if the numbers ...
Tyrcnex's user avatar
  • 572
0 votes
2 answers
59 views

Why must the population mean equal the population median if the density function is symmetric about the median?

Suppose we have $X_1,X_2,...X_n\overset{\text{iid}}{\sim}f(x-\mu)$ where, $f$ is symmetric about $0$. So, $\mu=$ median of $X$ and (if it exists) mean of $X$. Since, $f(x)$ is symmetric around $0$, $...
reyna's user avatar
  • 2,224
1 vote
1 answer
94 views

Finding the median value from a PDF

Suppose that the time in days until hospital discharge for a certain patient population follows a density $f(x) = (0.5)\exp(-x/2)$ for $x > 0$. What is the median discharge time in days? I reason ...
Stanley Yu's user avatar
0 votes
1 answer
64 views

Global minimum is attained at the median [duplicate]

I have the following homework problem. Let $(x_1,x_2,\ldots,x_n)$ be an increasing sample of size $n$. Show that the function $$f: x\mapsto \sum_{k=1}^n |x_k-x|$$ obtains a global minimum at the ...
modeltheory's user avatar

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