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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Please help me to understand how to read statistical tables

Sorry I never learnt from a professor or class how and now when I look at them I don't know what to do. Here is an example. The Chi Squared table, ...
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391 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
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27 views

26 flavours of ice-cream, how many different banana splits can be made that have 3 different flavours?

A boutique ice cream bar stocks 26 flavours and offers a rainbow banana split that contains 3 scoops of ice cream, each of a different flavour. How many different rainbow splits can the store ...
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40 views

Is the following a distribution function [closed]

Is the following a distribution function? $$F(x) =\begin{cases} e^{-1/x} &\text{ if } x>0\\ 0 &\text{ otherwise} \end{cases}$$ If so, give the corresponding density function. If not, ...
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1answer
16 views

Convolution for two random variables

In the textbook i'm currently reading it is said that for two independent random variables $X$ and $Y$ density function of variable $Z=X+Y$ can be found from the equation: $$ g(z) = ...
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27 views

Dealing with conditional probability

A programmer's programs are correct or buggy with equal probability $\frac{1}{2}$, independently. Three of his programs are randomly selected. Assume that one of the three is inspected at random, what ...
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10 views

Estimation effectiveness of two normal-distributed variables

You have two processes of measuring the air pollution, $X$ and $Y$. Both processes deliver values which are normal distributed around $\mu$: $X ~ N(\mu, \sigma_x^2)$ and $Y ~ N(\mu, \sigma_y^2)$. I ...
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9 views

Auto Correlation Function for AR(3) process

The AR(3) process is given by: $Y_k$ = $3\rho Y_{k-1}$ - $3\rho ^2Y_{k-2}$ + $\rho ^3Y_{k-3}$ + $C_0 W_k$ $W_k$ is a zero mean white noise whose variance is given by $E(W_k^2 )=\sigma ^2$. $\rho ...
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2answers
526 views

Poisson Distribution for Consecutive Figures

I am trying to find the probability for a Poisson distribution. The mean is two cars sold per day. The question is: "What is the probability that at least one car is sold for each of three ...
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1answer
25 views

Find the expected value of “Y”, exponential family with lots of questions here

I have a problem I don't know how to approach. It is A generalization of the 1-parameter exponential family, to allow 2-parameter distribution, is the family given by $$f(y;\theta, ...
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2answers
50 views

In how many ways can a dating service match each of six females one of eight males?

The question reads: A dating service has the names of six females and eight males who seek matches. In how many ways can the service match each female with one male? Using the multiplication ...
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15 views

Sub-Gaussian Random Variable with Small Variance

Write $X \in sG(\sigma^2)$ if $X$ is sub-Gaussian of parameter $\sigma^2$, that is $\mathbb{E}(e^{\lambda X}) \le e^{\lambda^2 \sigma^2 / 2}$. I'm interested in showing that, given $\epsilon > 0$, ...
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2k views

How do I find percentiles of data sets (Even vs odd)?

Given the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sample ...
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1answer
24 views

Understanding standardization for normal distribution

Let X be normally distributed random variable with expected value $\mu$ and standard deviation $\sigma$, then its СDF is: $$ F(x)=\frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^x ...
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16 views

Density Histogram interpretation in R

so I have this Histogram in R with the following data: ...
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23 views

If half of the UK's household wealth is owned by 10% of them, do we know anything about the mean?

A very simple question on maths: If half of the UK's household wealth is owned by 10% of them, is the cut off to that 10% the mean household wealth?
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29 views

Is the following modification of a martingale still martingale? [on hold]

I have a following question. Let $Z$ be a Geometric Brownian motion, $\frac{dZ(t)}{Z(t)} = \omega dt + \sigma dW(t) $ For $\omega = -\frac{1}{2}\sigma^{2}$ one can proof that $Z$ is a martingale. ...
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1answer
25 views

Asymptotic distribution of a measure of homogeneity

For an exam preparation I'm trying to solve the following question, but I get stuck. The question is One measure of the homogeneity of a multinomial population with $k$ cells and probabilities ...
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15 views

Someone verifies 500 light bulbs… - Method of moments and Maximum Likelihood

I have the following example: Someone verifies 500 light bulbs, there are bulbs with 0, 1, 2 or 3 errors. $X$ presentates the number of errors, $n_k$ presentates the frequency: \begin{array}{r|cccc} X ...
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4answers
52 views

What is the probability that there are $k$ people between $A$ and $B$?

If $n$ people are randomly seated in a row and two of the people are $A$ and $B$, what is the probability that there are $k$ people between $A$ and $B$ ($A$ can be either to the left or right of ...
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3answers
45 views

Intuition behid $P(A\mid B)$. [duplicate]

What is the intuition behind the formula $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$ I have seen this formula around, but every site/book I look at does not really have a clear & cut explanation behind ...
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37 views

Calculate the probability, that a man repair 20 machines in 8 hours. It is correct my work?

The problem statement said: The servicing of a machine requires two separate steps, with the time needed for the 1st step being an exponential random variable with mean 10 minutes and the ...
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2answers
16 views

Inverse of sum of matrices (SVD, ridge regression)

Looking at these slides, I've found the following: $X=UDV^T$, where $U$ and $V$ are orthogonal matrices, $V$ is a square matrix, and $D$ contains the singular values of $X$. The author then writes ...
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density of sum of two uniform random variables $[0,1]$

I am trying to understand an example from my textbook. Let's say $Z = X + Y$, where $X$ and $Y$ are uniform random variables with range $[0,1]$. Then the PDF is $$f(z) = \begin{cases} z & ...
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14 views

How can I caculate RMSE?

I have a project that uses Kriging. My original problems is actual values. I have some data about a region quality in 6 years but I don't know how to find actual values to calculate RMSE for choosing ...
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Of 100 people seated at a round table, more than half are women. Prove that there exist two women who are seated diametrically opposite each other. [closed]

Of 100 people seated at a round table, more than half are women. Prove that there exist two women who are seated diametrically opposite each other.
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2answers
407 views

How many strings in the letters a, b, and c have length 10 and exactly four a's?

part a) How many strings in the letters a, b, and c have length 10 and exactly four a's? I did $\binom{10}{4} = 210$ different ways for the strings of length $10$ in part a to be arranged, but ...
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2answers
50 views

What does $ X \sim (\mu, \Sigma)$ mean?

From textbook: We write for a random vector $X$ with mean vector $\mu$ and covariance matrix $\Sigma$, $$X \sim (\mu, \Sigma).$$ Does that mean (no pun) that $X$ is almost equal to a function of ...
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2answers
23 views

Correlation, dependence and the statement “Correlation does not imply Causation”

If we have two non-zero correlated random variables then they are dependent. Why then do we have the saying "Correlation does not imply Causation". A change in one variable may not cause exactly the ...
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1answer
368 views

Excel version of Skewed Normal Cumulative Distribution Function

I'm trying to deskew some distributions based on checks that verify Skew and Kurtosis and within range, if not, I was going to attempt to deskew the data. On wikipedia, there's a mathematical ...
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25 views

Mathematics Terminology

I was reading a paper, and the paper stated: $Cov_t (\epsilon_{a,t+1}, \epsilon_{b,t+1} \epsilon_{c,t+1}) =0$, for all $a$, $b$ and $c$. Does this mean that this also applies for cases where ...
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statistics of membership, a simple case

Suppose we have two lists of real numbers $S_1$, $S_2$ and a particular real number $x$, which we are not sure which group it belongs to. Are there any tests in statistics that predict the membership ...
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46 views

Rigorous Probability/Statistics Book reference?

Im wondering if anyone could recommend a book (or a few books) about statistics/probability for someone at the advanced undergraduate level who has taken some real analysis (at the level of baby ...
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1answer
25 views

Finding the conditional distribution of 2 dependent normal random variables

Here's the situation $X \sim N(\mu, \sigma^2)$ and given $X=x$, $Y \sim N(x, \tau^2)$ I need to find the distribution of $X$ given $Y=y$ From what's given, I know the pdf's of $X$ as well as ...
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22 views

Confused about Chebyshev and Empirical rule when applied to this question.

Here is the question I'm working on for my statistical methods class: Given a data set with a largest value of $760$ and a smallest value of $135$, what would you estimate the standard deviation to ...
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18 views

When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
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28 views

Finding Max Likelihood Estimators for the following Shifted Exponential PDF?

Consider a random sample $X_1, X_2, \dotsc, X_n$ from the shifted exponential pdf $$f(x;λ;θ) = \begin{cases}λe^{-λ(x-θ)} ;& x \geq θ\\ 0 ; &\text{Otherwise}\end{cases}$$ ...
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52 views

Probability Normal Disitrubution, $P(X <= 1.5)$

I think this is simple but I'm not sure how it works (I probably just dozed off while reading this part). The important info is that $X$ is a normal random variable with mean $3$ and standard ...
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18 views

The difference between $p$-value for $\chi^2$ with $k = 1$, and $\sqrt{\chi^2}$

Say you have a $\chi^2$ test statistic, with one degree of freedom. In order to find the $z$-score, $\alpha, \sigma$ significant (I'll just call $z$ in this post) of the alternative hypothesis from ...
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1answer
12 views

Statistic test for comparing two regression models

I'm having two linear regression models as follows: $y = a_1x_1 + a_2x_2 + c$ and $y = b_1x_1 + b_2x_3 + c$. I'm looking for a statistical test for proving which model is better. I've obtained the ...
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1answer
55 views

Never seen this notation before: $\int (y-f(x))^2 Pr(dx,dy) $

I have never seen an integral like this: $$\int (y-f(x))^2 Pr(dx,dy) $$ What is that? More precisely what is $Pr(dx,dy)$? And how is that integral defined? I found it in Elements of Statistical ...
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1answer
26 views

Why is Logistic Distribution called logistic?

What is logistic about Logistic Distribution, in a common sense way? What is the lexical rationale of the name, not just pure math definition?
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6 views

Data for “Seeking Shelter: Empirically Modeling Tax Shelters Using Financial Statement Information”

I wanna to reproduce this publication. But I can't find out the data. Do you know where is that publication's data? Seeking Shelter: Empirically Modeling Tax Shelters Using Financial Statement ...
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higher-order (3+) Taylor expansion of a likelihood function

I was wondering what is the effect if I replace the second derivative of the log-likelihood ("Likelihood" hereafter) function with its expectation in a higher-order Taylor expansion of the likelihood ...
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Why do we use nCk when determining numbers of favorable outcomes of coin tosses?

After delving back into probability a bit, I'm absolutely stumped as to why we would use nCk to answer the question "What is the probability of getting 3 heads when tossing a fair coin 10 times?" I ...
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31 views

How do I compute the following expectation for a matrix random variable?

I've successfully compute the expectation for the quadratic form of a random vector. But I've stumbled upon this matrix form. Let $X$ be a $n \times m$ random matrix and $A$ a $n \times n$ square ...
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22 views

Gradient of cost function

I have tried to calculate the gradient of the LMS cost function as follows but have a problem. $$J(\theta) = \frac12(y - X'\theta)^2$$ where $y$ is a scalar, theta and $X$ is a $n$ dimensional vector ...
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1answer
12 views

Converting F distribution to t distribution

Although I am aware of the relationship "square of the t distribution is F distribution", I'm having a little trouble applying it to the following problem: Use the t table to find the 80th percentile ...
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36 views

Confused on the relationship between Chi-square, its CDF, and p-value.

I have two questions on the $\chi^2$ statistic and $\chi^2$ distributions. I think I understand the $\chi^2$ test, in that for a given p-value, one can find a cut-off $\chi^2$ value for a given k ...
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2answers
25 views

cumulative distribution of intersection of events

Let $X_1,\dotsc,X_n$ be independent identically distributed random variables having common distribution function $F_X(\cdot)$. Express the event 'the smallest of the $X$s exceeds $k$' as an ...