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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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 ...
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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 < ...
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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?
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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 $n$, ...
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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.
$\int_{-\infty}^{m}f(x)dx = \frac{1}{2}$...
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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^...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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:
...
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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 ...
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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 $1=2$ ...
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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.
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Are median filters nonexpansive operators?
Let $m:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be the median filter associated with the family of windows defined by the rule
$$
W(i) = \{ j \in \{1,2, \dots, n\} : \max \{ 1,i-k\} \le j \le \min\{i+k,...
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Why must the population mean equal the population median if the density function is symmetric about the median?
Suppose we have $X_1,X_2,...X_n\overset{\text{iid}}{\sim}f(x-\mu)$ where, $f$ is symmetric about $0$. So, $\mu=$ median of $X$ and (if it exists) mean of $X$.
Since, $f(x)$ is symmetric around $0$, $...
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Combining two Left Skewed Distributions?
Let's say I have two groups of students, Group A & Group B.
We are going to test the groups with an examination paper, but on separate days. One group will complete the test on Monday, the other ...
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Finding the median value from a PDF
Suppose that the time in days until hospital discharge for a certain patient population follows a density $f(x) = (0.5)\exp(-x/2)$ for $x > 0$. What is the median discharge time in days?
I reason ...
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Global minimum is attained at the median [duplicate]
I have the following homework problem.
Let $(x_1,x_2,\ldots,x_n)$ be an increasing sample of size $n$. Show that the function $$f: x\mapsto \sum_{k=1}^n |x_k-x|$$ obtains a global minimum at the ...
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How to Find the Median of a Probability Distribution Function?
I am interested in learning about how to find out the Median (https://en.wikipedia.org/wiki/Median) for some "generic" Probability Distribution Function (https://en.wikipedia.org/wiki/...
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Calculating list size from original and altered average when x is removed. [closed]
The average of x number of values is 66, and when a certain 77 is removed from these values, the average drops to 62. Find x, or the number of values that when averaged give 66, and when 77 is removed ...
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Median of medians algorithm [closed]
My question is I don't understand how we determine the constant in T(T(n/3), T(n/5)). For example, we divide elements into 9 groups and get a formula like this
T(n) <= T(n/9) + T(7n/9) + O(n)
I don'...
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Area of a triangle $\triangle ABC$
Let be $\triangle ABC$ a triangle with its area of $48\text{cm}^{2}$. $M$ is the mid of $[BC]$ and $P$ is the mid of $AM$. We know that $BP \cap AC = \left\{N\right\}$.
The question is: how to compute ...
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Given a set of many 2d lines, computing the median line for multiple x positions?
I have a set of 2-dimensional lines, defined by $y = a_i x + b_i$ for $i$ in range 1 to n.
I also have a separate set of query x positions $x_j$ for j in range 1 to m.
I need to compute the median ...
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Convergence of median and convergence in distribution
Let $X_n$ be a sequence of real random variables such that $X_n \rightarrow^{d} X$, that is, $X_n$ converges in distribution to $X$. Suppose that $X$ has a cummulative distribution function, $F$, ...
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Prove that $\text{P}(\max\limits_{1\leq k\leq n}|X_k-m_k|\geq t)\leq 2\text{P}(\max\limits_{1\leq k\leq n}|X_k-X_k^{'}|\geq t)$
Let $\{X_k:1\leq k\leq n\}$ and $\{X_k^{'}:1\leq k\leq n\}$ be two independent sequences of random variables (i.e the two sequences are independent,but the variables in the same sequence are not ...
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By comparing boxplots, how to tell which sample is more likely to have outliers?
For example, from three boxplots, like these
how could I tell which one is more likely to have outliers?
By outliers, I mean values below $Q1 - 1.5 * (Q3-Q1)$ or above $Q3 + 1.5 * (Q3-Q1)$
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Median and Mode from grouped frequencies
I'm currently learning about grouped frequencies and I was wondering how we could determine the Median and Mode from grouped frequencies. We don't know the precise values so we can only determine and ...
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Is the median of CDFs again a CDF?
I am working on the following barycenter problem: Suppose we are given $N$ probability measures on $\mathbb{R}$ with cumulative distribution functions $F_1,\dots,F_N$ and we are interested in the ...
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Generalization of mean and median
It is well known that a median of a distribution $\mu$ can be defined as an $m$ such that
$$m\in\operatorname*{arg\,min}_{c\in\mathbb{R}}\mathbb{E}_{X\sim\mu}[|X-c|].$$
Similarly, the mean of a ...
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Efficient way to verify if pair of numbers are medians of partition of sets
Given a multi-set of multi-sets $\mathcal{S} = \{S_1, \ldots,S_i, \ldots, S_n \}, S_i \subset \mathbb{R}$. Denote powerset of $\mathcal{S}$ as $\mathbb{P}(\mathcal{S})$.
My question is, given a pair $(...
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How to think about changes to the median
I've been asked the following question, and, while it is obviously pretty simple, I've got myself in a tangle overthinking it. The question is:
"Imagine that you have data from a web server, ...
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Assuming a random variable X has a strictly decreasing pdf, prove that its mean is strictly greater than its median
Got asked this question in an interview. More generally, I would like to understand for which distributions is the mean greater than the median and vice versa. For example, it is known that for ...
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$o_p(1)$ property of sample median estimator
$\textbf{Motivation:}$ Van der Vaart - Asymptotic statistics [Example 5.11]
$\textbf{Setup:}$ Suppose I have $i.i.d$ random variables $X_1,...,X_n$. Define the sample median $\hat\theta_n(\omega)$ as ...
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How to prove that the median of a set cannot surpass a given value by contradiction?
I hope you're having a good day, I'm very new to statistics, and I'm having trouble with this question:
"The average age of 100 people is 30 years.
Prove by contradiction that the median of the ...
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Median values in probability distribution where two median values can possibly occur?
I have a relatively basic question, I will keep it short, sharp and shiny.
In a probability distribution lets suppose $P_1 =0.5, P_2 = 0.0, P_3 =0.5$
All three outcomes share a slice of the median ...
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How do I find the median of this continous random variable?
The question given was to find the median of $X$ given $f(x)=1.1e^{-1.1x}$
Random variable is $X≥0$
I have no issue working out the median of probability density functions with parameters like $1≤X≤5$ ...
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Can a median value occur twice?
From a practice exam:
From the set $$\{7, 12, 5, 16, 23, 44, 18, 9, Z\},$$ which of the following values $Z$ be equal to if it is the median of the set?
Choose from: $14, 11, 12, 17, 21.$
$14$ was ...
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Bounds on $k$-medians
For a metric space $(X,d)$, given $N$ points $S=(s_1,\dotsc,s_N) \subset X^N$, a $k$-median is a multiset of centers $K=\{c_1,\dotsc,c_k\} \subset X$, $k\leq N$, that minimizes
$$
f_S(K) \equiv \sum_{...
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