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Questions tagged [means]

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. For a data set, refers to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.

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Calculate conditional probability using mean and variance

I have a set where its values follow a normal distribution, but I only have the sum of all of them, and the sum of its squares. Having: $\sum_{i=0}^n X_i$ and $\sum_{i=0}^n X_i^2$. I have the ...
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Correlation of means or correlation of individual values

I have a data set of player performances and am looking to calculate some correlations of their stats with win percentage. Some players have multiple performances, is it valid for me to take the mean ...
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How to understand that the solution to least squares problem transformed with Box-Cox Transformation, is a generalized mean with $h(x)=x^\lambda$?

The least squares problem $\min_a \sum_i^n (x_i-a)^2$ is sometimes solved using transformed variables, that is, solving $\min_a \sum_i^n [h(x_i)-h(a)]^2$. The solution to this latter problem is $a=h^{-...
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I need help in this statistics problem. Can somebody help?

So I have this raw data for a case study: Note: 1 stands for average weight and 2 for overweight. I need to calculate the following: What is the mean expected time spent for the average-weight ...
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Mean preserving spread for normal distribution

Is it true that any two normal distributions with the same mean can be ordered w.r.t. the relation of a mean preserving spread? My intuition would be that this is true but I cannot come up with a ...
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number of times an experiment has to be repeated for optimal mean

I am trying to get the total execution time of a program - running it different times will give different run times since it depends on a lot of factors. Thus, I am running this program multiple times ...
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Calculate weighted mean based on simple mean

I need to calculate the mobile weighted mean of a quantity, $a_i$, based on $b_i$. I have a tool that allows me to calculate the mobile mean of a given series, so we can say I have a function: $F(...
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I can't think how to approach this statistics problem?

A set of numbers is transformed by taking the log base 10 of each number. The mean of the transformed data is 1.65. What is the geometric mean of the untransformed data? ...
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Mean and Median are equal. Find x and Y. X and Y are natural numbers. [closed]

4,6,x,y,10. The mean and median are equal to each other. Find x and y?
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Hypothesis testing for sample means within a normal distrubution

The context of the question is that a bakery bakes cakes and the mass of cake is demoted by $X$ such that $X \sim N(300, 40^2)$. A sample of 12 cakes is taken and the mean of the sample is 292g. The ...
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Calculating the Mean of a Range with Floor and Ceiling Functions?

I did a survey a couple months back, and one of the questions required a range of numbers. I may have discovered my own formula for how to calculate the mean of a range of numbers, but I don't know ...
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Riemman-Stieltjes Integration-IVT

I need to show this, Suppose that $f,g: [a,b] \rightarrow \mathbb{R}$ are continous. Show that exist $\eta \in (a,b)$ such that $$g(\eta)\int_a^\eta f(x)dx=f(\eta)\int_\eta^b g(x)dx $$ I defined $...
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Are the mean and variance of a set equals the sum of its means and variaces

I'm not so fit in statistics and I found some controverse answers on the internet so I'm asking here. I have a set $A$ with 10439 samples. The set is not of unique values and many of the sample ...
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What is the probability mass function of the measured voltage?

In a specific design, the true voltage of a circuit is 250 millivolt(mV). Measurement error that is continuous and uniformly distributed from -3 to +3 mV is added to the true voltage. the measurement ...
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should I use the mean or weighted mean in this example?

An investor calculated the percentages of each of three stock investments with payoffs as shown. Find the average payoff for the three stock investments. what should I use in this case to calculate ...
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Finding variance without the data

A researcher is testing the effectiveness of a political video. She has randomly sampled 120 people to test the video. For each person, there is 50% chance the person will be Democrat and 50% chance ...
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Mean and Variance of total number of fishes.

The number of fishes caught by a skilled fisherman each day has a Poisson distribution with mean $2.5$. After $50$ days, what is the mean and variance for the total number of fishes the fisherman ...
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$Ε[Χ]$ calculations with different distribution variables

I have just started my stochastic processes course this semester and I came across this problem. I am missing something for sure because the $X_s$ confuses me. Let $A,B$ two independent random ...
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Interpolation between “arithmetic mean” and “geometric mean”

Define a function $f(X; p)$, which measures a statistic on a set of points $X = {x_1, ..., x_n}$ (and $x_i \in (0, 1)$), with respect to a parameter $p$. I am looking for ways to define the function $...
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Limit of generalized mean with exponent tend to zero

How to get the following limit: $$\lim_{p \to 0}\sqrt[p]{\frac{1}{n}\sum_{i=1}^{n}x_i^p}=\sqrt[n]{\prod_{i=1}^n x_i}$$ I.e. how to get geometric mean from definition of generalized means? I attempted ...
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Compute the correlation coefficient $r(X_{(1)},X_{(3)})$ .

I got this problem where: The random variables $X_1, X_2,$ and $X_3$ are independent and $Exp(1)-$ distributed. Compute the correlation coefficient $r(X_{(1)},X_{(3)})$ . I know through research ...
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Question about an inequality about geometric mean

I encountered this inequality when reading some paper, however, why it holds does not seem very obvious to me: Consider nonnegative real numbers $a_1, ..., a_n, b_1, ..., b_n$, and $b_i \leq \frac{...
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Variance and mean of balls in bins limited capacity

Let there be $m$ indistinguishable balls, $k$ bins, $C$ capacity. Let $X_j$ denote the total balls in bin $j$. I've seen ways to calculate the total number of combinations, but I'm not sure how to go ...
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Expected value of number of throws of a dice to get element $1$ four times

Here is the question: Find the expected value and variance of the number of times one must throw a dice until the element $1$ has been obtained $4$ times. My attempt: The minimum number of throws ...
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MSE Minimized by Mean

In the Gaussian case, it is well-known that the MSE, is minimizer by the mean value. However, in general, if $X \in L^2(\mathcal{F};\mathbb{P})$, is a random-variable in $\mathbb{R}$, then is the ...
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How to calculate Measurement error of n measurements based on standard error and the quantile funcion

Let's say I have $n$ measurements $x$ sampled from a normal distribution X and I would like to estimate the mean with errors. Is the belove method correct? I'm mostly concerned about if I'm wording ...
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What is the maximum value of $\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$?

If $x$ is positive, what is the maximum value of this expression: $$\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$$ This question is from a book of problems on sequence and series under the section on AM-...
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Can a summation be transferred into the denominator?

I include a bit of an introduction, even though my main question is more mathematical. I was tasked with finding the Maximum Likelihood Estimate for $\theta$ in $$\mathrm P(X>x) = \left(\frac ax \...
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Is a definite integral just a summation?

I am learning about definite integrals and found the formula for finding an average of a function over a given interval: $$\frac{1}{b-a} \int_{a}^{b} f\left(x\right) dx$$ If we look at the average ...
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How to show that the number of jobs is distributed equally

Sorry if I'm asking an obvious question. I have proposed a load balancing algorithm for a system of several devices. Suppose at first the number of tasks on each device is $device1 = 18$, $device2 = 8$...
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Mean of the partial sum of the normalized Gaussian random vector

Let $X_1, X_2, \dots, X_d$ be $d$ independent $N(0, 1)$ random variables, and let $Y=\frac{1}{\|X\|}X$, where $X=(X_1, X_2, \dots, X_d)$. Let the vector $Z \in \mathbf{R}^{k}$ be the projection of $Y$ ...
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Consider a random variable $X$ with the log-normal pdf $f(x) ={1\over \sqrt{2π}}x^{−1}exp^{{−0.5 (logx)^2}}$, $x>0$.

Consider a random variable $X$ with the log-normal pdf $f(x) ={1\over \sqrt{2π}}x^{−1}exp^{{−0.5 (logx)^2}}$, $x>0$. a) Find the mean and the variance of $X$. I know that $\int_0^\infty f(x) dx=1$...
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Is there a formula for the silver spiral?

I can construct a golden spiral by rotating segments 90 degrees and scaling their segments by the golden ratio, and then use the polar formula on wikipedia to interpolate points at other angles to ...
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My goal is to calculate $\operatorname{Cov}(X,Y)$, but I'm struggling to calculate $E[XY]$

Regarding the following problem: A fair coin is tossed 5 times. Let $X$ be the number of heads in all $5$ tosses, and $Y$ be the number of heads in the first $4$ tosses. What is $\operatorname{Cov}(...
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Computing the overall mean value in a game of chance with related means

Suppose you are presented with the following game: Game Interface How the game works: Player places bets on one or more symbols. The wheel is spun. If the wheel lands on a symbol that the player ...
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How should I calculate mean CI - from raw data or mean values?

I have experimenatal data from medical industry. In the experiment 4 observations were weighted, each one 2 times for more precise measurement. As a final weight, a mean from each pair is reported. ...
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Is there an upper bound for the mean of a ratio between two random variables?

Is there an upper bound for the mean of a ratio between two random variables ?
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Standardising a score of values

I am working on developing on a standardised score from a number of scores. The values will be ranging from large to small of each score and I am trying to develop a way of proportionally measuring ...
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Best Average to Represent Quantitative Data?

For a final assignment, my two group members and I have created a biased survey called Should Public Schools Implement a Uniform Policy? We have about 2-3 ...
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Do two pairs of distinct natural numbers exist such that AGM(A,B) equal to AGM(C,D)?

Here AGM is arithmetic-geometric mean. Are there natural numbers A,B,C,D such that $1\leq A<C<D<B$ and arithmetic-geometric mean AGM(A,B)=AGM(C,D) ? In other words, is AGM a homomorphism ...
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Can arithmetic-geometric mean of two distinct natural numbers be a natural number?

Are there solutions to equation $AGM(A,B)=C$ such that $1\leq A<B$ and $A,B,C\in\mathbb{N}$?
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Mean vector and covariance matrix

I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that? Given: $$\begin{equation} p_\underline x(...
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Sampling uncertainty (uncertainty in standard deviation)

Say I took twenty measurements of the wind velocity at a point. This is a sample of the infinite number of measurements I would need to get the true velocity at that point. I then calculate the mean ...
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What is the mathematical meaning of this question?

$a,b,c \in\mathbb{Z}$ and $x\in\mathbb{R}$, then the following expression is always true: $$(x-a)(x-6)+3=(x+b)(x+c)$$ Find the sum of all possible values of $b$. A) $-8$ B) $-12$ ...
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Range of Data Values That Includes 95% of my Data

For a math assignment, I am required to find the range of data values that would include 95% of my data. My data is the wait time for a specific bus. I have gathered 20 pieces of data for this, ...
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Generalized AM-GM Inequality

I was discussing means with my friend, and I tried to illustrate the concept of geometric mean using the following idea: Suppose we have two positive quantities $x,y>0$. The simplest geometric ...
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Statistical problem: Scores, Weights and Averages

As part of a software development project, I'm trying to find the correct mathematical approach for a statistics problem involving scores, weights and averages. Here's a simplified summary of the ...
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Hypothesis Testing for Probability and Statistics

Question: The present defective rate of a product in a certain factory is 3%. A random sample of 250 of products produced using a new technique which is supposed to improve the quality of production ...
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Generalized definition of all means of $n$ numbers?

Following chapter 2 of Greek Means and the arithmetic-geometric mean by George and Silvia Toader, the Greek (or Pythagorean) means of two numbers have been studied for centuries originating in the ...
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What questions lead to different representative statistics?

What are the questions that are asked whose answer yields the geometric and harmonic means. For example, we arrive at the arithmetic mean by solving: $$ \min_y \sum_{x \in S} (x - y)^2$$ What ...