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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Establishing the upper and lower bounds of normal using standard deviation

I understand the concept of standard deviations and z-values, but I'm trying to figure out if standard deviations alone are good for establishing the upper and lower bounds for normal. For example, if ...
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23 views

Generalization of method of least squares to matrix system (Pseudo inverse?)

A and B are two m$\times$n real matrices with m > n. I need to find X: a real m$\times$ m matrix such that $\| A - X B\|$ is minimized. On thing I'm thinking about is using the singular value ...
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410 views

Variance of a MLE $\sigma^2$ estimator; how to calculate

Let $X_1, X_2,...,X_n$ be an i.i.d. random sample from $N(0, \sigma^{2})$. a. Find the variance of $\hat{\sigma}^{2}_{MLE}$ So I found $\hat{\sigma}^{2}_{MLE}$ by taking the derivative of the log ...
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The sample distribution (pdf) of the sample mean retrieved from gamma distribution

Is it true that the sample distribution (pdf) of the mean where sample is of size n retrieved from a gamma distribution with shape a and scale b is given by ...
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DRIFT MATRIX in Ornstein Uhlenbeck Process

The Weiner Process was unable to explain Brownian Motion and then there was the need of Ornstein-Uhlenbeck Process. The Ornstein-Uhlenbeck Process describes the Brownian Motion in the presence of ...
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Indicator variables and instrumental variables

Consider that we have a problem of endogeneity in the classical linear regression model $\operatorname{cov}(x,u)\neq0$. We find an instrument for this endogenous variable. Suppose the instrument is ...
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Sampling distribution of $Y = \frac{\ln U_1}{\ln U_1 + \ln (1 - U_2)}$, where $U_i \sim U(0,1), \forall i$

For this problem I have used the fact, $-2 \ln U \sim \chi^2_{(2)}$. But I have doubt on the independence of numerator and the denominator which are $\ln U_1$ and $\ln U_1 + \ln (1 - U_2)$. If they ...
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34 views

About 'Marcinkiewicz–Zygmund inequality'

Marcinkiewicz–Zygmund inequality gives gives relations between moments of a collection of independent random variables. The statement of this inequality can be seen in Wiki ...
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306 views

Numerical calculation of fisher information

I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ ...
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28 views

how manys ways are there if the order is taken into account?

Three candidates are selected from a certain number of interviewess. if the order is not taken into account, the number of ways the candidates can be chosen is 35. how manys ways are there if the ...
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$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}}$ , where $X \sim N(0,1)$

Prove that: $$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}},$$ where $X \sim N(0,1)$ and $c>0$, $t \in\mathbb R$. The problem should be solved easily by using the equality: $$P(X \geq c) = P(e^{Xt} ...
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51 views

Probability: bakery distributes pies

I'm working through a mathematical statistics textbook, and I can't get a question right. It is a follow-up to this question: At the end of the day, a bakery gives everything that is unsold to ...
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25 views

How to distribute a cost in a normal distribution

I need to spread out a number so that it reflects a normal distribution. For example, I have an item that cost $500,000$ dollars in year $2050$ and I would like to spread it across with a standard ...
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1answer
436 views

Weighted Standard Deviation for Histogram Bin Height

I'm plotting some binned data in the form of a histogram. Say I have 10 data points, each composed of a bin to be placed in, and then a "height". Then I might have something like: Bin Height 0 - ...
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41 views

probability of threshold crossing [closed]

Let $\{X_i\}$, $i=1,2 \ldots$ be $\textit{i.i.d.}$ positive random variable distributed as (some) $F(\cdot)$ with finite mean. Let $S_n= X_1+ X_2+ \ldots+X_n$ be the sum of $n$ $X's$ and let $a>0$ ...
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292 views

expectation calculation in probability and statistics

2 four-sided dice are rolled. X = number of odd dice Y = number of even dice Z = number of dice showing 1 or 2 So each of X, Y, Z only takes on the values 0, 1, 2. (a) joint p.m.f. of (X,Y)? joint ...
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23 views

Determining and excluding the outliers of a dataset

I am trying to model biological processes using Ordinary Differential Equations. I have a (pretty large) model that I am trying to parameterize using software (Copasi's implementation of the Genetic ...
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Bounds on Chi Squared Distribution

Consider the following hypothesis test: $\mathbf{X} =(X_1,\cdots, X_k) \sim $Multinomial$(n,\mathbf{p})$ and $H_0 : \mathbf{p} = \mathbf{p}_0 = (p_1,\cdots, p_k)$. I know to test this, we construct ...
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Instrumental Variables and orthogonality conditions

To obtain the method of moment estimate for instrument variables, we use the moment condition $z'\varepsilon=0$ in the exact identified case (number of endogenous variables = number of instruments) or ...
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26 views

Leave K out cross validation shortcut

For splines and linear regressions there is this handy shortcut: Let $\hat{f}$ be a spline estimate of a true function $f$, and let $\hat{f}_i^{[-i]}$ be the model fitted to all data except $y_i$. ...
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Prove that $E (\overline{X} - \mu)^2 = \frac{1}{n}\sigma^2$

How to prove $E (\overline{X} - \mu)^2 = \frac{1}{n}\sigma^2$ (from wiki), where $\overline{X}$ - is the sample mean ? What I have so far: \begin{align} E (\overline{X} - \mu)^2 = ...
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calculate $E(X^2),E(X^4),\ldots$ for various random variables

Is there any document or tool directly showing the results of $E(X^2),E(X^4),\ldots$ for various random variables? where $E$ is the expectation and $X$ is a kind of random variable that may follow ...
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With the constraint $E(X)=0,E(X^2)=1$, is Rademacher (symmetric Bernoulli) variable X the best choice to minimize $E(X^4)$?

Rademacher variable $X$ means that $X$ can be either $-1$ or $1$ with equal probability $0.5$. Then my question is that: Is Rademacher (i.e. symmetric Bernoulli) variable X the best choice to ...
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Measure of how well a 3D function represents experimental data

I have experimental data of a one dimensional heat equation and corresponding values for a predicted temperatures. Is there any method in which I could statistically analyse the data to determine if ...
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Statistical Significance of a Simple Test

Please help me with this basic question on statistics: If a standard brick is dropped on a standard raw chicken egg from 1 meter; the egg breaks. How many times does this dropped-brick-onto-egg need ...
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44 views

Find probability using geometric distribution

I wanted someone to check the solution of this problem (see Rice's book, problem 2.14) Two boys play basketball in the following way. They take turns shooting and stop when a basket is made. Player A ...
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83 views

Intuition behind odds of winning 0 times approaching 1/e

I am learning about the number $e$. The wikipedia page says that in a Bernoulli experiment the odds of never winning in $n$ trials approach $1/e$ as $n$ tends to infinity. I am trying to develop an ...
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How to find CDF and PDF of $Y = 4X(1-X)$, given $X\in[0,1]$ [closed]

Let $X$ be uniform on $[0,1]$ and $Y = 4X(1− X)$. Find the CDF and PDF of $Y$.
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In terms of $a, b,$ and $\theta$, what is the biased $b(\hat \theta)$?

The Statement of the Problem: Let $\{P_{\theta}: \theta \in \Theta \}$ be a statistical model. Suppose that $\hat \theta$ is an estimator for a parameter $\theta$ and $E_{\theta}(\hat \theta) = ...
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Name for a constrained Poisson-like bridge process

I have a sequence $t_i$ for $i=0,2,\cdots,n$ of integer jump times with $t_0=0$ and $t_n=n$ such that the waiting time $t_{i+1}-t_i$ has distribution density $f_i(t)$. So it's kind of like a Poisson ...
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If $X$~$U(0,1)$, and $Y=2x-4$. What is the density function of Y?

If $X$ is uniformly distributed $\mathcal{U}(0,1)$ , then what is the distribute density function of $Y$? I thought that if $$fx(x) = 1/(1-0), \; \mbox{for} \; 0<x<1$$ then ...
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comparing percentile ranks of two normal distributions with 1 std difference in medians

We have normal distributions A and B. Distribution B's median is 1 standard deviation to the right of distribution A. What percentile in Distribution A is 98th percentile of distribution B?
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50 views

Probability of multiple dice rolls with decreasing amounts of dice

Calculating probabilities over multiple dice rolls is easy, but what do you do if the amount of dice decreases (dependently) from roll to roll? This is a common feature of many games, including Risk, ...
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51 views

Show that $X_n = n \min(T_1,T_2 \cdots T_n )$ has assymtoticaly an exponential distribution as $n \rightarrow \infty$

Let, $T_1,T_2 \cdots T_n $ be i.i.d random variables having reliability function: $R-(t) = 1 - \lambda t - o(t)$ as $t \rightarrow 0$. Show that $X_n = n \min(T_1,T_2 \cdots T_n )$ has ...
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How can mathematical models be applied to image analysis

I'm quite interested in how mathematical models can be used in analysing images. For example, I'm aware that mixed effect models can be using in image analysis but I was just wondering if there are ...
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Math Factorials. Simplifying by distrubution. I am confused.

Say we are working with statistics and factorials. In the proof of ... $$\frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!(n-[n-r])!}$$ How is $(n-r)!(n-[n-r])!$ supposed to distribute to the simplified ...
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335 views

Proof for Standard Deviation Formula for a Binomial Distribution

I understand the concept of standard deviation as the square root of the square of the mean of each sample value - the mean of the sample values. Here is the mathematical representation (I've solved ...
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How does accuracy of a survey depend on sample size and population size?

Which survey is more accurate? Assume the samples are taken perfectly randomly. A sample of 100 people out of a population of 1000 (sample is 10% of population) A sample of 1000 people out of a ...
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On the central limit theorem

The Central Limit Theorem states for a sequence of i.i.d. random variables $\{X_i\}$, $$\frac{\overline{X} - \mu}{\sigma/\sqrt{n}} \to N(0,1)$$ in distribution as $n \to \infty$. I saw in some ...
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60 views

Bootstrap method failing where blocking works

I'm computing an average of individual samples that are not entirely independent and need an estimate for the true standard deviation. According to Newman and Barkema's book the most reliable method ...
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43 views

A Question about the difference of Two samples

Problem: From each of two normal populations with identical means and with standard deviations of $6.40$ and $7.20$, independent random samples of $64$ observations are drawn. Find the probability ...
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Probability of inequality with Chi-squared distributed random variables

I want to evaluate analytically the following probability: $P(X+XY\leq Z+ZW)$ where $X\thicksim \chi_1^2$, $Y\thicksim \chi_a^2$, $Z\thicksim \chi_1^2$, and $W\thicksim \chi_b^2$ with $a,b\geq 2$. ...
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Mathematical Intuition behind the tf-idf formula in Statistics

I was reading: https://en.wikipedia.org/wiki/Tf%E2%80%93idf#Definition But I cannot seem to understand exactly why the formula was constructed the way it is. What I do Understand: iDF should at ...
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Finding new standard deviation and mean after adding an element

Say I have a mean and standard deviation for a dataset of 5 elements. I now add a sixth element. Is there a way to calculate the new mean and standard deviation using the information we had prior ...
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Maximum and minimum Expected values when taking colored balls

We have a sack with $60$ balls. From them $15$ balls are red, $15$ green, $15$ blue and $15$ yellow. We take $30$ balls from the sack. What's the expected number of balls of the color from which ...
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Statistical independence of degree in Erdos-Renyi random graph model

Let $d(v)$ denote the degree of the vertex $v$ in the random graph $G$ coming from the Erdos-Renyi model. I would like to calculate $\mathbb{E}[d(v) d(u)]$. Clearly, $$\mathbb{E}[d(u)] = ...
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1answer
19 views

Comparing percentages of a sample to that of the population.

This might be stupid question, but I'm in this sort of situation: 60% of people in a city have a pet cat, but the national rate is 50%. So, assuming we have the required bits of information about ...
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36 views

Using central limit theorem for normal distribution

So i have the following question: The times that patients spend in a doctor’s surgery have mean 5 minutes, and standard deviation 2 minutes. On one particular day, the doctor sees 30 patients during ...
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Weighted Jaccard Index

i have a litle problem trying to apply Jaccard index between sets that comes from diferent populations. A little example can better show my problem: We can imagine the 'human jobs ontology' that ...
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Normalizing relative list of probabilities

I have an array of objects, and I want to randomly select one. These objects all have a performance property that ranges between [0, 1]. If this performance value is greater than (or equal to) some ...