Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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Conditions for an “Algebra” of probabilities

Suppose I have events $A,B,C,D,\ldots$ and that my probability space is discrete and finite (As nice as possible). Now, suppose I give you the values of say $P(A,B)$ , $P(A|B,C)$, $P(D, B|A)$ etc etc. ...
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1k views

ELO ranking system: what's a good start rank?

I'm looking to implement an ELO ranking system. I've read the wikipedia articles and I'm confused about the start rank for players who enter the system at a later point. The common solution is to use ...
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933 views

Computing the probability that at least four telephone lines are not in use.

A mail-order computer business has six telephone lines. Let $X$ denote the number of lines in use at a specified time. Suppose the pmf of $X$ is as given in the accompanying table: $p(0)=.10$, ...
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65 views

Comparison between the calculated average value and the probability q of a Bernoulli random variable

I'm using the R language to generate a sample of 32000 independent values ​​Bernoulli (q) with q = 0.1 to construct a dot plot with the respective averages calculated using the first N values ...
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193 views

Gamma Densities proof question

I understand the first portion of the problem but then how do I conclude that $f_n(x)$ gives us a probability density function on $[0,\infty)$. If someone could point me in the right direction that ...
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166 views

Verifying Exponential Family

Why is the following $$f(x|\theta) = \theta^{-1} \exp(1-(x/\theta)), \ \ 0 < \theta < x <\infty$$ not an exponential family? We know that $$f(x| \theta) = h(x)c(\theta) \exp(w(\theta)t(x))$$ ...
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123 views

Nomenclature of random variables $\{X=0, Y=0\}$ same as $\{X=0\}\cap \{Y=0\}$?

just a small doubt. My exercises keep oscillating their nomenclature on this small detail and I always have the other version. Let $X,Y$ be random variables. Is $\{X=0, Y=0\}$ the same as ...
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481 views

Observed information matrix is a consistent estimator of the expected information matrix?

I am trying to prove that the observed information matrix evaluated at the weakly consistent maximum likelihood estimator (MLE), is a weakly consistent estimator of the expected information matrix. ...
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116 views

Exponential Families

A family of pdfs or pmfs is called an exponential family if $$f(x|\theta) = h(x)c(\theta) \exp \left(\sum_{i=1}^{k} w_{i}(\theta) t_{i}(x) \right)$$ What is the motivation of this definition? It ...
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493 views

Intuition of Gamma Family

The function $$f(t) = \frac{t^{\alpha-1}e^{-t}}{\Gamma(\alpha)}, \ \ 0 < t < \infty$$ is a pdf. But Why is the gamma family defined as $$f(x| \alpha, \beta) = \frac{1}{\Gamma(\alpha) ...
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593 views

offspring distribution

At time 0, a cell culture starts with one red cell. At the end of one minute, the red cell dies and is replaced by 2 red cells with probability $\frac{1}{4}$, with 1 red and 1 white cell with ...
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330 views

Order Statistics Expectation and Variance

How would we go about finding the expectation and variance of the r$^{\text{th}}$ order statistic $X_{(r)}$ from a random sample $X_1, \ldots, X_n$ from a uniform distribution with density function ...
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200 views

Might such a sequence of mathematical expectations be able to predict uncertain events?

This question might sound a little bit mystical, but it seemed like an interesting idea, so I am posting it here. Despite the title, I know it probably does not work miracles, but here goes anyway. I ...
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1answer
257 views

Incorrect use of Borel Cantelli?

I am confronted with the following argument which I think may not be right: Let $(X_n)$ be a sequence of independent random variables s.t. $$P[X_n = 1] = 1- P[X_n = 0 ] = \frac{1}{n}$$ in order ...
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198 views

Birth-death process invariant distribution

Let $X_n$ be a birth-death process, with birth rates $\lambda_n$ and death rates $\mu_n$ (with $\mu_o=0$ and $\lambda_{-1}=0$). How do you show that the invariant distribution $\pi_i$ is: ...
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265 views

iid Random Variables

Let $X_i \quad$ be independent identically distributed Random Variables s.t. $E|X|^q < \infty$ ( some $q \in \mathbb{N} )$. When defining $Y_i:= (X_i-\bar{X}_n)^q$ why is it true that $Y_i$ are ...
2
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261 views

Maximum Likelihood Estimation

For iid random variables from a distribution with p.d.f. $$f(x;\theta_1,\theta_2)=\frac{1}{\theta_2}\exp\bigg(-\frac{(x-\theta_1)}{\theta_2}\bigg), \quad x>\theta_1, ...
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9k views

Average percent increase not equal to total percent increase?

I tried searching around for this but it was difficult to boil down the search terms. Plus nothing seemed to be showing up anyway. What's an easy way to show that the average percentage increase of n ...
2
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3answers
91 views

z-interval and sample size what is a normal sample size

Can a Z-interval be used when the sample size is between 15-30? does the variable play a role? I'm not too sure if it makes a difference. I know it can be used if the population is a normal or large ...
2
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2answers
992 views

Showing a distribution is not complete for a parameter $\theta$

I am trying to show that a normal distribution with parameters $\mu = 0$ and variance $\theta$ is not complete. I am looking for a function $u(x)$ that is not equal to 0 such that $\mathbb E(u(x)) = ...
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76 views

Simultaneous equations for bacteria.

I would like to calculate the value of bacteria on 4 surfaces $i=\{1..4\}$. A person touches some of those 4 surfaces at random and a count is made on their finger after each surface contact ($x_i$). ...
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158 views

A theorem about inductive inference

In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read: 'Inductive inference is well known to be a hazardous ...
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3answers
380 views

Residuals of regression model

Let's suppose I do a regression between earnings and age (and suppose I do not know the distribution of earnings). Would it be possible for the residuals to be normally distributed? I am thinking it ...
2
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1answer
173 views

Hypothesis test (p-value)

Suppose I conduct a hypothesis test and $H_0: \mu = 7$ and $H_A: \mu \neq 7$. Let's say the $p$-value is 0.03. Would a 95% confidence interval contain the value $7$? I think it would since alpha ...
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51 views

What are problems an $M$ - estimator is trying to solve?

Anyone here have any experience with $M$ - estimators and do you think you can give a brief explanation that the problem an $M$ - estimator is trying to solve ? Thanks.
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185 views

Create 'smooth breakpoint function' by using integral?

Experts, I am a biologist and thus my natural strength is not math, yet I´m quite okay with statistics. Now I am facing the problem that I have to find an unusual (?) mathematical solution for a ...
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1answer
873 views

Is it possible to compute the variance without computing the mean first?

I have a list of values of a random variable $x \in \mathbb R$. Is it possible to find the varience $\overline{(x - \overline x)^2}$ without computing the mean $\overline x$ first? That is to process ...
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1answer
55 views

Distribution from quantile data or custom distribution

I'd like to fit a distribution (any you like) based on these requirements: Produces integer values (preferable but not required) Mean ($\mu$)=100 Std=114 Quantiles( 25%, 50%, 75%)=(6,39,200) Min=0; ...
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465 views
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474 views

Conditional probability. Targeting events

Electric motors coming off two assembly lines are pooled for storage in a common stockroom, and the room contains an equal number of motors from each line. Motors are periodically sampled from that ...
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2answers
308 views

probability density function

Let us say that $X$ is a normal random variable, given that its expected value is $0$ and variance is $1$ . How do we compute the density function of $Y= e^ X$? What I think: Since expected value is ...
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1answer
7k views

What does it mean to divide by the standard deviation?

I'm trying to variance normalise an image, in order to do so, I subtract the mean from the pixel value, to have a 0 mean, and divide by the standard deviation, to have a unit variance, right ? but ...
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134 views

Showing completeness

How would I show that the order statistic Y5 is complete for the parameter theta? I have the pdf of Y5 but I am unsure of the process on how to show completeness using the definition. Any help would ...
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2k views

Is there a sufficient statistic for shifted exponential distribution?

Suppose we have $X_1, \ldots, X_n$ i.i.d, $X_i \sim Exp(1, \mu)$ (pdf is $f_\mu(x) = e^{-(x-\mu)}$ for $x \geq \mu$ and $0$ for $x < \mu$). Is there any one dimensional (i.e. $T: \mathbb{R}^n \to ...
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171 views

differentiating “n” question

i would like to know the difference here (this is no longer a homework question but rather concept differentiation) we know that probability = s/n (where s = success and n = total outcomes) in this ...
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66 views

how many ways? permutations/combinations

suppose i have this set { BLUE, BLUE, GREEN} how many ways can i arrange this, given that {Blue 1, Blue 2, Green} is the same as {Blue 2,Blue 1, Green} i cant find a reason to call it ...
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1answer
124 views

solving combination problem

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439 views

Use Pearson's correlation coefficient on a matrix

I have a problem to interpret the following formula which is said to be the Pearson's correlation coefficient: $$r = \frac{N \left(\sum XY\right) - \left(\sum X\right) \left(\sum ...
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2answers
3k views

If two Gaussian random variables are uncorrelated, they are statistically independent

I read in a textbook that when two gaussian variables are uncorrelated, then they are statistically independent? How can I prove that?
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460 views

Q-Test in mathematica

I want to use the Q-test to determine outlier in my dataset. However, I cannot find a function to do this is in Mathematica. Does anyone know if there is a function which does this in Mathematica? ...
2
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1answer
2k views

How to efficiently estimate quantile function of Gamma distribution

I have an application that analyzes datasets comprised mostly of samples from a Gamma distribution. Mixed in with the data are an unknown number ($>= 0$) of outlier samples (which are actually taken ...
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55 views

Sampling a distribution with restrictions: eliminating the correlation between two variables

I have a collection of $400.000+$ word-pairs. Each word-pair has an association strength, which is a measure of how related the two words are to each other (as in cow-milk). Each word-pair also has a ...
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2answers
2k views

Inter-arrival time distribution

It is known that for a Poisson process the inter-arrival time is exponentially distributed. My question, which may be nonsense, is this. Suppose you want to experimentally evaluate the distribution of ...
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1answer
88 views

In general, $\textit{Var}(X) = E(X^2) - [E(X)]^2$

This is from page 25 of this book: In general, it may be shown that $$\textit{Var}(X) = E(X^2) - [E(X)]^2$$ I can't remember ever seeing that "In general" elsewhere. So if this identity only ...
3
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1answer
260 views

How does one handle ties in Order Statistics and Rank-Order Statistics?

I'm preparing to calculate a data set's Mean Difference (equation given in the image below), but am having trouble understanding what I'm supposed to do when duplicate values are involved. My data ...
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241 views

A simple question about sampling variance [duplicate]

Possible Duplicate: A question about sampling distribution We defind a variance of a random variable $X$ as $E(X-X_{\text{mean}})^2$=$E(X^2)-(E(X))^2$ and the mean = $E(X)$. However, for ...
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3answers
361 views

Derivation of pmf from convolution

Suppose that a discrete random variable (with finite support) $Y$ is given by $Y = X_1 - X_2$, where $X_1$ and $X_2$ are both discrete random variables with finite support and with the same ...
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100 views

Combining multiple posterior distributions

I am new to Bayesian statistics, and thus have problems to come up with a solution for the following problem: Using Approximate Bayesian Computation (ABC), I generate a posterior distribution from ...
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434 views

What is to median as first central moment is to mean?

The question sounds like a riddle, but it isn't intended to be one. I've been thinking about the Cauchy Distribution which, famously doesn't have any central moments defined. A very informal ...
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2answers
2k views

Generalization of variance to random vectors

Let $X$ be a random variable. Then its variance (dispersion) is defined as $D(X)=E((X-E(X))^2)$. As I understand it, this is supposed to be a measure of how far off from the average we should expect ...