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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Use the Central Limit Theorem to find the value that makes an expression true.

Suppose you roll a standard die 2000 times and let X be the sum of the values you get. Using the Central Limit Theorem, for what value of a is P[X >= a] approximately equal to P[N(0,1) >= 2] ?
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1k views

Prove an Unbiased Estimator of p.

Suppose you have a random variable X with distribution Geom(p), and you would like to estimate p. Since E(x)=1/p, a natural approach to estimate p is to get a sample from X and then take the ...
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148 views

Find $P(X+Y>1)$ probability of density function

Yet another PDF question. I am given the following joint density function, $$f_{X,Y}(x,y)=\frac{2}{5}\text{ when }0<y<1\text{ and } 0<x<5y$$ And have to calculate the probability that ...
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48 views

Variance of $Y^2$ of density function

I am working with some density functions and got stuck when asked to find the variance. I currently have the following, $$f_Y(y)=2y\text{ when } 0<y<1$$ $$E[Y]=\frac{2}{3}$$ I can calculate ...
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1answer
119 views

Detecting periodicity in point processes

I have data from a periodic one dimensional point process representing the times of events which is also mixed in with lots of data that I regard as noise. The total number of points is of order one ...
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81 views

Multilinear or Tensor Regression?

Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
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1answer
222 views

find the cumulative distribution function

Suppose we have a joint probability density function $f(x,y)$, how do we find the cumulative distribution function of $ T = X + Y $ if the support is $$ \begin{cases} 0 < X < 1 \\ 0 < Y < ...
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84 views

Finding parameters for curve fitting

I have 500 observed data of variable $ x $ and corresponding $ y $. The functional model is where Is it possible to find suitable constants $ A , B $ ,$ \alpha , \beta $ so that the observed ...
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121 views

Expected number of trials before first success

I have a question which I hope statistics experts on this forum can help me with. Given a time series data D(t) with mean m and median M what is the expected time (number of trials) before value y is ...
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1answer
386 views

find marginal probability density function

Suppose random variables X, Y have joint probability density function $f(x, y)$. How do i find the marginal probability density function of X , Y if the support is $$ \begin{cases} 0 < x < 1 \\ ...
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621 views

What is the Probability that a Knight stays on chessboard after N hops?

Say a $8 \times 8$ chessboard as per picture. A position is represented here by co-ordinates $(x,y)$. A move is aslo considered as valid, where the Knight lands outside the chessboard [ For eg. ...
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154 views

Methods for determining thresholds for future use based on historical data (in R or Exce;)

Let's say I have some data about the amounts of reports to Help Desk (about technical problems) which were monitored and registered every five minutes for the whole day (even during the night). Is ...
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35 views

Uniform choice for Prior Distribution

My prior function is $\Phi\left(\mathbf{k}_\ell,W_\ell\right)=\frac{1}{N}\log p\left(\mathbf{k}_\ell,W_\ell\right)$ which is determined once I choose the Bayesian prior parameter likelihood ...
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37 views

Interaction effect in ANOVA .

The statistical model for a Randomized Complete Block Design (RCBD) with multiple observations per cell is $$y_{ijk}=\mu+\tau_{i}+\beta_{j}+(\tau\beta)_{ij}+\epsilon_{ijk},\quad i=1,\ldots ,a, \quad ...
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3answers
2k views

Kendall tau calculation

Can someone explain how the Kendall tau works? I can't seem to find a good explaination/tutorial/example. I've been running corr(x,y,'kendall') from Matlab's ...
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2answers
314 views

What's the probability that an NFL team with a given win/loss record makes the playoffs?

For example, if I know that a team has a record of 11 wins and 5 losses, but no nothing about the records of any other teams, what is the probability that this team makes the playoffs? The current ...
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1answer
66 views

Regarding calculating the bias of coin with uncertainty

Suppose you have a coin that you flip $n$ times and the result have $m$ heads and $n-m$ tails. How accurate can you predict the bias of the coin to be $\frac m n$? I know that ...
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Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$. But what about the law of the iterated logarithm? Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely (from ...
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1answer
95 views

Calculating success chance from algorithm

Not super sure this is the right *exchange for this question, but here we go. Let's say I'm writing a game, and in this game the player may attack another unit. The chance of hitting is an "opposed ...
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1answer
193 views

Regarding probability bound of flip coins

Suppose you flip a fair coin 10,000 time how can you characterize the distribution of the occurrence of head? From the textbook, it says that $P[head>\frac{n}2 + k\sqrt{n}]$ < $e^{-{k^2/2}}$, ...
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14 views

How to find the best fit when you have a set of ideal ratios, but some of those are below a minimum?

Say you have a set of ideal ratios, whose sum = 1. For example, input = [0.2, 0.2, 0.3, 0.3] But suppose that there is a rule stating that every ratio should be ...
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187 views

Is the MLE strongly consistent and asymptotically efficient for exponential families?

It is known that the Maximum Likelihood Estimator (MLE) is strongly consistent and asymptotically efficient under certain regularity conditions. By strongly consistent I mean that $\hat{\theta}_{MLE} ...
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53 views

Simulating a Bernoulli experiment with a very large numbers of trials. each trial represents a single bacteria mutating or not mutating

I am helping a friend of mine in making a simulation in MatLab involving the mutation of bacteria. First of all I would like to apologize in advance for my lack of knowledge of statistics. There is ...
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55 views

Joint PDF for spherical region

A sphere has a coordinate system (r, $\theta$, $\phi$) with the origin at the center of the sphere. What is the joint PDF of the r and $\phi$ coordinates, $f_{r,\phi}(r,\phi)$, for a randomly ...
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SLLN of Markov chains .

Let $X_1$, $X_2$,... be a finite state, irreducible and aperiodic Markov chain with initial state $X_0=i$. It is known that \begin{equation} \mathrm{P}\Big\{\lim_{n\rightarrow\infty} ...
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1answer
247 views

confidence intervals - a bad confidence interval

I had a question for confidence intervals: the situation in the question :so we have a number of scatter plots with each showing an estimated regression line (based on a valid model) and associated ...
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50 views

Hermite rank of an $L^2$ function

Let $(H_k)_{k\in\mathbb{N}}$ be the sequence of hermite polynimials, $Z\sim N(0,1)$ and $G\in L^2(\mathbb{R},\phi)$ with $\operatorname{E}\left[Z\right]=0$. By $\phi$ we denote the density of the ...
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1answer
22 views

Normal distribution finding P area

Q: let X be a continuous random variable with NORMAL DENSITY $$f(μ,σ(x)) = \frac{1}{\sqrt{2}π}*σ *e^{−(x−μ)^2/ 2σ^2}$$ We know that $μ = 70$ and $σ = 2$. Find $P(68 \leq X \leq 74)$ and $P(X \geq ...
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26 views

Minimization values for a function [duplicate]

I have got function - non linear(I thk), and a set of variable S=[(x1,y1),(x2,y2)...]. The objective is to find the value for ...
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1answer
375 views

Statistics: Why do we divide by $\sqrt{n}$ for sample standard deviation

Can someone tell me if my explanations/understanding is on the right track? Suppose we have a set of variances, each of them identical, where $V_{1}(x) + V_{2}(x) + ...+ V_{j}(x) = \sigma^2$. If we ...
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97 views

Probability of Specific event occuring between 2 events?

Forgive me beforehand for what may be a question with an obvious seolution, but I havent had statistics courses in quite some time. I have an Excel File of approximately 3000 Events, each event has a ...
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41 views

Chi square proof with Exp law

For an exercice in Statistics, For $ X \sim Exp(\theta)$ I have to proof that : $ (\dfrac{2} {\theta}) \sum_{i=1}^{12} X_i \sim X^2(24)$ And with this result, I have to found a best critic ...
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1answer
24 views

data interpretation

let us consider following picture and question: INCOME: 1. Money from fund raising programmes, 2. Grant from the government, 3. Contributions from individuals, 4. Contributions from corporations, ...
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34 views

Expected number of draws to draw a unique item given outcome probability distribution with replacement

Given a mystery box with multiple outcomes, if the outcome of concern has a probability of being drawn with 15%, what is the expected number of draws to draw this outcome? Can we assign a dollar ...
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2answers
705 views

Where did this statistics formula come from: $E[X^2] = \mu^2 + \sigma^2$

I am studying statistics and I need some guidance as to where this formula came from. All I know is that $\displaystyle E[X^2] = x^2 \sum_{i=0}^n p_{i}(x)$
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1answer
218 views

(theoretical) Negative Binomial Distribution using Matlab

I was trying to solve some exercises on Matlab in order to improve my skills and I stumbled upon this question: For the (theoretical) Negative Binomial distribution with parameters r = 5, p = 0.4, ...
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39 views

Adapt 1 to 40 scoring matrix using scoring-variable survey feedback on 1 to 10 scale

We have a scoring instrumented consisting of 7 variables with various arbitrary weights of up to 40 for ea. variable (Scales noted below). To add objectivity to our method, we conducted a ...
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29 views

Kendall Tau distance over rankings containing different elements

Suppose to have two lists (or rankings) containing the same number of elements (but not the same elements). E.g.: $[5, 4, 3]$ vs. $[5, 4, 2]$ Dow do you define the Kendall tau distance between the ...
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223 views

Distribution of the $l_2$-norm of gaussian vector

Let $Y_k \sim N(\mu_k, \sigma_k^2)$. For $\sigma_k = \sigma$ the squared norm of $Y = (Y_1, \ldots, Y_n)$ follows the noncentral chi square distribution. What is the distribution in the general case? ...
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51 views

Why the definition of Variance is such. [duplicate]

Why we define the variance of a random variable $X$ as $\text{var}[X]=\text{E}[(X-\mu)^2]$ instead of $\text{var}[X]=\text{E}[\left|X-\mu\right|]$. Normally we understand the standard deviation ...
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48 views

Statistical Significance level - elementary level

Let's say one day my sister tells me she has psychic powers and can help me predict the winners in horse racing games and for whatever reason, we only have 2 horse racers in a game. She tells me that ...
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2answers
47 views

Number of times you have to make a bet on a uniform distribution to expect to achieve a minimal result

Edited for the sake of clarity: If you have a random variable $Q$ distributed uniformly on some interval, say $[a,b]$, what is the function $f$ that describes how many times you have to draw on the ...
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70 views

Sum of two log-normal

Let X and Y be two GBM’s, they have each a univariate log-normal distribution for some time t, that is $X_t\sim{LnN(µ_x, σ^2_x)}$, $Y_t\sim{LnN(µ_y, σ^2_y})$ and $Z_t=[X_t,Y_t]\sim{ MvLnN(μ, Σ)}$ ...
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impossibility of Axiom 3 of probability

Suppose $S = [0,1]$ and $P$ is a uniform probability measure. Find a countable collection of subset of $S$, $A_1, A_2, A_3,\dots$ such that they are pairwise disjoint but $P(\cup A_i) \neq \sum ...
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1answer
196 views

The largest of $N$ random numbers over a uniform distribution?

So I read somewhere than if you have $N$ numbers picked independently from a uniform distribution, say $[0,1]$, the greatest number has an expected value of $\frac{N}{N+1}$. So if you have 2 numbers ...
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1answer
251 views

Weighted Average of Percentage Increases

I believe I have quite a simple problem, but want clarification on whether it is the best method to use. Say I have have insurance line, and the net income (£) from this business for 2011, 2012, 2013 ...
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1answer
26 views

Trying to understand an equation for predicting baseball, specifically a comma?

I'm reading a paper about predicting baseball games, and I'm having trouble figuring out what one of the variables means. It's the "r1, r2, and r3" variables in the following passage: It is ...
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2answers
135 views

Does convergence in probability not imply convergence in distribution for Least Squares estimators?

I have a question relating to convergence in probability and distribution for least squares estimators. Frequently, I see in textbooks that $\hat{\beta} \rightarrow^p b$. Where $b$ is the population ...
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2answers
35 views

Looking for a function that gives more weight to more “data points”, i.e. 30/60 > 1/2

I'm looking to calculate "conversion rates" of some items, as follows: (number of times the item was clicked on) / (number of times the item was presented to the user) I want to give a higher ...
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How $\mathbb E[\bar\epsilon_{i.}-\bar\epsilon_{..}]=0$ ? $\mathbb E$ denotes expectation.

Statistical model for Complete Randomized design $y_{ij} = \mu + \tau_i + \epsilon_{ij}$ where, $i$ denotes treatment and $j$ denotes observation. $i=1,2,...,k\quad and \quad j=1,2,..., n_i$ ...