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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Using Chebychev Inequality to show a distribution stochastically approaches zero

Let $X_1,X_2, \dots$ be independent Bernoulli random variables, $X_i \sim BIN(1,p_i)$ and let $$Y_n=\sum\limits_{i=1}^n (X_i-p_i)/n.$$ Show, using Chebychev inequality, that the sequence $Y_1, Y_2, ...
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35 views

Is the following a distribution function [on hold]

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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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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24 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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13 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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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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37 views

Discrete Time Markov Chain question

Let $\{X_n : n \ge 0 \}$ be a Markov chain with state space $ \{0, 1, 2, 3\} $ and transition matrix $$P=\begin{pmatrix} \frac{1}{4} & 0 & \frac{1}{2} & \frac{1}{4}\\ 0 & \frac{1}{5} ...
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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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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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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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4answers
49 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 ...
3
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3answers
44 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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1answer
24 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
42 views
+50

Conceptual Statistics. Define for this problem, population, Samples and Estimators, and when is Normal Dist?

Students in Stanford are supposed to spend on average 3 hours of time per week for every credit hour they take. Last year, 263 randomly selected seniors were contacted and asked how much total time ...
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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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Are standard error and absolute error the same? [closed]

I want to calculate mean absolute error and mean bias error . My data are measured and predicted values and standard error.Are standard error and absolute error the same?
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3answers
53 views

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
49 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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0answers
21 views

can it be proven, that manufactoring x products perfectly is impossible? [closed]

I don't know if this question is silly or not - if it is, sorry for asking, I'll delete my question then. I'm curious if the impossibility of manufactoring a certain amount of products without any ...
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0answers
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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23 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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2answers
44 views

Computing Probabilities [closed]

I am new to statistics, and I just came across this question, I am a little unsure how to solve it.. X and Y are two students who take a graduate course in probability. The course has only three ...
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2answers
23 views

Statistics in circuits

I have a circuit, in order for it to work the electricity has to travel from point A to point B, but on its way there it encounters 4 independent switches. If any one of them is closed the system will ...
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1answer
20 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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1answer
21 views

statistics question relating to population [closed]

the questions is there are $27 \%$ pure bred dogs, and the rest are mixed breed. $35 \%$ of all dogs are aggressive and its known that $70 \%$ of purebred dogs are aggressive. my question is how do I ...
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1answer
24 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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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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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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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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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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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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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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8 views

Help with Stats Asymptotic Efficiency [closed]

Question I can find the variance of T2, using the delta method, but I'm unsure how to calculate T1. Any help would be greatly appreciated. Thanks!
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Relation between average of z values and average of probabilities for standart normal random variables? [closed]

If $Z_{1},\ldots,Z_{k},\ldots ,Z_{K} \sim N(0,1)$ and $a_1,\ldots,a_k,\ldots a_K$ are real numbers with average $a$, $P_k=\Pr\{Z_k>a_k\}$ for all $k\in \{1,\ldots,K\}$ with average $P$ what is the ...
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2answers
49 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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1answer
10 views

What is the population when conducting a study for the amount of television watched and the level of physical fitness amoung adult males? [closed]

Would the population be all adult males or all adult males who watch television?
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1answer
30 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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1answer
21 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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2answers
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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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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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4answers
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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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1answer
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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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0answers
16 views

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 ...
2
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2answers
24 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 ...
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binomial distribution elementary quesiton

A car repair garage has two spare sets of spark plugs in stock for a particular model of car. Each time one of these cars comes in for repair, there is a probability of $0.05$ (independently on ...
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Probability in a Dice Game [duplicate]

By the following problem: Two players, $A$ and $B$, play a game where they, one after another, throw two dices. The first one to get a sum of $A$ wins the game. Let player $A$ be the one to play ...