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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Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
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Learning the geometry of spaces of probability distribution as fast as possible

Strange statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence or max/minimization with respect to these quantities are ...
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Limiting Distribution $\Delta-$method

Let $Y_n\sim \chi^2(n)$. What is the limiting distribution of $U_n= \dfrac{\sqrt{Y_n}-\sqrt{n}}{\sqrt{2}}?$. What I know is that if $X_i\sim \chi^2(1)$, I can write $Y_n = \sum\limits_{i=1}^n X_i$. ...
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Finding the MLE for an open interval.

So the problem says: Let $X = (X_{1},...,X_{n})$ be a random sample, where $X_{i} \sim Unif (0, \theta _{0})$, where $\theta _{0} \in (0,\infty)$ is unknown. Find the maximum likelihood estimator $T$ ...
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1answer
39 views

How to find $z$-score

I have some probabilities, but I have to find the $z$-score. I am not sure how do to this when I am told I have to use slope-intercept. Where do I plug the numbers in exactly? Here is one of my ...
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Likelihood function for a distribution with both discrete and continuous compoents

Suppose $X_1, X_2, \ldots, X_n$ are $IID$ normal RVs with mean $\mu$ and variance $1$. However, we observe only $Y_i$'s where $Y_i = \max (0, X_i)$. I would like to know how to write likelihood ...
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If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator?

If $X\sim\operatorname{Poisson}(u)$ and $\theta = \mathbb{P}\{X=0\} = e^{-u}$, is $\hat{\theta}_1 = e^{-X}$ an unbiased estimator? Here's what I tried, is this right? $$ \begin{align} ...
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An applied example of Moment Generating Function in R? Or other software

I'm trying to understand MGF, I get the theory but I'd like to find an example I can relate using a software, like R studio or Matlab. Any example of one would be really appreciated!
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17 views

Median of Medians

Given a set A with median Am = 10 and set B with median Bm = 20 is it true that the median of the combined set C is $10 \le$ Cm$\le 20$ ? My first thought was that this wasn't true so I tried to find ...
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Estimate the number of “solvable” polynomials with degree $d$ and coefficient limit $l$

Enumerate the irreducible polynomials with degree $d$ and integer coefficients $[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the ...
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How to find variance of weighted sum in terms of inputs and weights ? Variance of inputs and weights are given. [on hold]

Let $X = [x_1, x_2, ... , x_M ] $ and $ W = [w_1, w_2, ... , w_N] $ $y_i = {\sum_{j=1}^N}{\sum_{i=1}^{M} {x_iw_j}}$ Variance of $X$ and $W$ are given. Can we find the variance of $Y$ in terms of ...
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1answer
25 views

If $Y = (\mathcal{N}(\mu_1,\sigma_1^2) + \mathcal{N}(\mu_2,\sigma_2^2))^2$, what is $\Pr(Y>\mathrm{E}[Y])$?

Given $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, with $X_1$ independent of $X_2$, as well as $Y = (X_1 + X_2)^2$, what is $\Pr(Y>\mathrm{E}[Y])$? ...
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1answer
22 views

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ , what is the mean of $X$?

The density of a random variable $X$ is $f(x)$ proportional to $x^{-1/2}$ for $x \in [0,1]$$ and $f(x) = 0$ for $x \notin [0,1]$. Then, the mean of $X$ is $\frac 12$ $\frac 1{\sqrt2}$ $\frac 13$ ...
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Obtaining probability density function $f_Y(y)$ when we know joint probability distribution $f(x,y) = 1/(x+1)$

Suppose joint probability density function is $f(x,y) = 1/(x+1)$ for $0<x<1$ and $0<y<x+1$. I try to calculate marginal density function $f_Y(y)$ by $$f_Y(y) = \int_{y-1}^1 ...
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40 views

Why the Sum of all possible outcomes does not equal to one, in this case?

The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution. Edit:- ...
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Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$.

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer? $r_1 = r_2$. $r_1 = 10r_2$ ...
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probablitly using bayes theroem

Three numbered urns contain colored balls as described in the table below. One of the urns is picked at random and a ball is drawn from the urn; the ball is red. What is the probability the ball can ...
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Which distribution would be the most appropriate?

What standard distribution would be suitable for the random phenomenon at hand, and what are the knowns and unknowns? e) The size of an automobile insurance claim I'm thinking that the distribution ...
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24 views

An explanation of how this solution is derived

I am having difficulty understanding the solution to this problem. Since the solution is in the form of Bayes theorem I expected something along the lines that looked similar to Bayes theorem. ...
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Likelihood Function for the Uniform Density. $ (\theta-1,\theta+1)$

Let the random variables $X_1,X_2,...,X_n$ iid $U[\theta-1\,,\theta+1]$. So the likelihood function therefore has the form: $L(\theta|X)=\prod_{i=1}^nf(X_i|\theta)=\frac{1}{2^n}I(X_1, . . . , X_n ...
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Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...
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19 views

Minimum of four exponential variables

Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents? Attempt: ...
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48 views

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$.

Suppose $P(|X| < 1) = 1$ and $P(|Y| = 2) = 1$. Then which of the following is true? The standard deviation of $X$ is smaller than that of $Y$. The mean of $X$ is smaller than that of $Y$. The ...
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How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf ...
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Prove that if $X_n \xrightarrow{P} c$, then $E(X_n) \to c$ for $X_n$ uniformly bounded

I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that $$E\left(X_n \right) \to c$$ ...
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How to find expectation of birth-death process

Let $\{X(t)\}$ be birth-death process on two-state space {0,1}. Let birth rate $\lambda=2$ and death rate $\mu=12$. How to calculate $\lim_{t\to\infty} \mathbb{E}[X(t)]$?
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Proof of the DKW inequality

My goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) : Let $(X_i)_{i \geqslant}$ be iid random variables. Let $\displaystyle F_n(x)= ...
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Calculate the estimators of $E[X]$ and $Var[X]$ using the method of moments

$(X_1,\dots, X_n)$ is a random sample extracted from a uniform distribution on the interval $$(\alpha-\beta, \alpha+\beta) \ \ \ \ \alpha \in \mathbb{R}, \beta \in \mathbb{R}^{+}$$ Demonstrate ...
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In the context of ordered statistics, each of Y(1),Y(2),…,Y(n) a single observation or distributions that are I.I.D?

In statistics one aspect of the I.I.D. concept that bothers is when I think about it in the context of ordered statistics. As most of you already know, $Y_1,Y_2,Y_3,...,Y_n$ are I.I.D. when the ...
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Please help me create a mathematical model that can be used to compare two different strategies [on hold]

I have been pondering this question: "Is it better to be balanced, or focus on one area?" Let's say that I can decide how to divide my investment between two products. I want to know if it is better ...
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statistics help [on hold]

A charity solicits donations by phone. From long experience the charity’s director reports that 60 percent of the calls will result in refusal to donate, 30 percent will request more information via ...
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If $P(A) = 0.2$, $P(A \cap B) = 0.1$, $P[(A \cup B)'] = 0.3$, what is $P[(A \cap B) \mid (A \cup B)']$?

Suppose events $A$ and $B$ are such that $P(A \cap B)= 0.1$ and $P[(A \cup B)'] = 0.3$. If $P(A)=0.2$, what is $P[(A \cap B) \mid (A \cup B)']$? I tried solving it by using the conditional ...
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Covariance/Correlation Proof

I'm having a little problem with a statistics problem I am working on. I'm not really sure where to start to prove the two statements. Any help would be greatly appreciated. Let $x$ and $y$ be ...
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Notation Bayesian Statistics $\propto$

I often read the following notation: $\propto$. How is this sign called and what is the definition of it?
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How to analyse data set

I have a data set from work I'm not sure how to analyse. We have a bunch of people we've contacted (about $1,000$) at a random time during a project they're conducting. That time is expressed as a ...
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44 views

The mathematics of Correlation is not equal to Causation

In statistics, it is a common practice to say that "correlation does not mean causation", and mostly the proof for this is given by examples. While that is good for the intuition, it's not rigorous. ...
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Proposal Criteria Analysis

We've just finished evaluating three proposals on the following criteria: Criterion Point A1 125 A2 125 A3 100 A4 150 Cost 500 ----- ----- Total ...
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Sufficient statistics 2

Given $X_1, ..., X_n \ iid$ from a family of distributions parametrised with $\theta$, $Y = g(\underline{X})$ sufficient statistic for $\theta$, $Z = h(\underline{X})$, that $Y$ and $Z$ are ...
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Analysis of calls to a call center using poisson distribution

I have a set of data from my workplace where we note how many support calls we receive. I have been playing around with it in my spare time just to see if I could predict anything interesting. I have ...
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Moment Estimator

I have been given Independent sample variables $X_1,...X_n$ that have a common p.d.f $$f(x,\theta)=\frac{10\theta^2}{x^2},$$ where $0<\theta<x$. How do I go about in finding the moment ...
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Statistics - Mean of Ungrouped Data

Two weeks before James opened technology titans he launched his company web site. During those 14 days James had an average of 24.5 hits on his Web site per day. In the first two days that technology ...
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Find the expected value of the matrix

$\require{cancel}$ I want to see if I have solved this problem appropriately or not. If we have ...
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proof for the equivariance of the MLE

I am self-learning statistics and I read about the theorem that under some conditions the MLE is equivariance. I couldn't find any proof for that theorem. What are the conditions and what is the ...
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31 views

What does the capital $E$ notation followed by curly bracket mean?

While reading through a statistics book earlier today I came across a notation I'm unfamiliar with and can't find a way to search for it. It is not expected value $E[\,]$, but instead the following. ...
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1answer
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Why was poisson distribution introduced?

I am studying probabilites and the notion of poisson random variable was introduced in the class. But it seems to me that the introduction of poisson random variable is to provide a easy approximation ...
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Finding MLE for $\mu^{2}$

The problem says the following: Let $X = (X_{1}, ..., X_{n})$ be a random sample, where $X_{i} \sim N(\mu_{0},1)$, where $\mu_{0} \in \mathbb{R}$ is unknown. I do not have problems calculating the ...
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1answer
42 views

Obtaining quadratic equation using Least Squares Method

This question is most likely extremely trivial, but I'm having some difficulty obtaining the least squares equation from the following data points: {{1.08, 0}, {1.07, 0.0659232}, {0.97, 0.1695168}, ...
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U-statistics and Independent Sum

I have i.i.d. paired samples (X,Y): $(X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n)$ I compute the statistics $\sum_{i \neq j} X_i \cdot Y_j$ People have told me that the above is actually a sum of $n$ ...