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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Computing $E[ {\rm Tr}\{(ZZ^T)^2 \}]$ for $Z$ Gaussian.

Let $Z \in \mathbb{R}^n$ be a Gaussian random vector with zero mean and $Cov(Z)=I$ where $I$ is identity matrix. How to compute \begin{align*} E[ {\rm Tr}\{(ZZ^T)^2 \}] \end{align*} I know that ...
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25 views

How do I find middle range with normal distribution?

I am currentLy reading my textbook there is an example that I cant understand. Can someone explain to me the hint given to solve it? Q. Verbal SAT scores follow normal(430,100) distribution. What is ...
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19 views

Statistics and data analysis

What is meant by confidence interval in data analysis e.g. 95% confidence interval? How does p<0.05 estimate significant difference?
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52 views

1 scratch off, 1 ticket or both? [closed]

I have a problem that I would like some input on. Free beer(or soda) for best answer. I have several scratch offs and several tickets options. I have the maximum loss(item price) and I have the ...
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23 views

Check if the weak law of large numbers holds true for the following sequence of random variables

Suppose we have $n$ independent discrete random variables, whose distribution is as follows: $X(k)$, where $k$ is any integer from $1$ to $n$, can take any of three values: $-\sqrt{k}$ with a ...
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17 views

distribution function from a sample

I have a sample about relative error of a measuring device and I wanna know the error probability of that device. What non parametric test must I use to know its error probability distribution of the ...
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32 views

What is the probability density function of **the multiplication of Gaussian variables**?

Assuming $x_1,x_2,\ldots, x_n$ are $n$ independent variables from standard Gaussian distribution $N(0,1)$. Then we construct a new variable by $y=\Pi_{i=1}^n x_i$. Can anyone show the probability ...
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41 views

Combining normal distrubutions

I am not sure of the terminology here, if this is a product, summation, or average. How can you take a two unimodal normal distributions and combine them into a bimodal distribution? And then combine ...
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5 girls and 3 boys are arranged randomly in a row. Find the probability that…

5 girls and 3 boys are arranged randomly in a row. Find the probability that: a) the 5 girls are next to each other,= 2/28 b) the 3 boys are next to each other,=3/28 c) there is one boy on each end, = ...
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49 views

Which math class can I take to learn how to derive statistical models

I have taken several stats classes and. Have seen many models in action like the normal, poisson, dirchet, etc. and seen several inference tests in action like chisq, ttest and anova. However I'm ...
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36 views

Difference between two proportions in a Confidence Interval

Ten engineering schools in the United States were surveyed. The sample contained $250$ electrical engineers, $80$ being women; $175$ chemical engineers, $40$ being women. Compute a $90\%$ confidence ...
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Monopoly Game Statistics

I was playing a game of monopoly the other day, and in the course of strategizing I came up with the idea that how 'safe' you were in the game was a matter of what your expected income/outcome was as ...
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How to prove: $E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$?

How to prove: $$E(|tr(x^Tww^Ty)|^k)\leq \|yx^T\|_2^k E(tr(x^Tww^Tx)^k)$$, where $k$ is a positive integer, $x,y$ are fixed vectors, each entry in $w$ i.i.d. follows from an standard norm ...
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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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1answer
15 views

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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12 views

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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139 views

Why are these following variance and expected value computations legitimate?

I spent over an hour of my exam's given time to calculate the variances and expected values as given here: Let $p,q\in (0,1)$. The number of costumers entering a supermarket is a r.v. $X$ with ...
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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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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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35 views

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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41 views

$P(X \geq c) \leq e^{-ct +\frac{t^2}{2}}$ , 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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3answers
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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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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20 views

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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15 views

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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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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35 views

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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32 views

Estimating the number of classes in a finite population [closed]

Suppose I have N smarties, each of which is one of C distinct colours. Suppose further that N is known and largish (10,000) but C is not, and that for each colour C there are $c_i$ smarties of that ...
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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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1answer
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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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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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22 views

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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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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1answer
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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1answer
14 views

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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32 views

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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3answers
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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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1answer
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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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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33 views

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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2answers
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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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88 views

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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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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33 views

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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1answer
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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1answer
22 views

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 ...