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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Statistics: Adding up averages

Let's assume we the average income for a journalist is $10000$ (in the whole country). In a certain state X the average income for this profession is $10700$, therefore 7% higher. Also, the average ...
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Bound of diameter in Erdos-Renyi

I would like to compute the bound of the diameter in random graph $G(N,p)$ following Erdos-Renyi model. Anyone can tell me how to compute this bound? Thank you so much.
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Fitting a model to a collection of binomial proportions, based on varying (large) sample sizes.

I have a multi-parameter bivariate function, say $f(i,j)$ that I want to use to predict the entries of a matrix $M(i,j)$, the entries of which are binomial probabilities based on varying sample sizes, ...
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9 views

linear transformation of factorial moment generating function

Let $X$ be a discrete random variable with factorial moment generating function $\psi_X(t)$ and define $Y=aX+b$, where $a$ and $b$ are constants. Express the factorial moment generating ...
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19 views

Probability of reception from n users

Say I have 4 boxes on the ground. At every time interval the nth user(n = 1,2,3...) has some probability of throwing 0, 1, 2, 3, or 4 balls into the boxes. Each box can only hold one ball, and the ...
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Statistics - Probability

QUESTION If a new type of torch battery has a voltage that is outside certain limits, that battery is characterised as a failure (F); if the battery has a voltage within the prescribed limits, it is ...
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20 views

Degree of nodes in Erdos Renyi model

I see in the E-R model with a random graph $G(N,p)$ on $N$ nodes and probability of edge existence $p$, the probability that a node has degree $d$ is $$ P(d)=\binom{N-1}{d} p^d (1-p)^{N-1-d}$$ Give a ...
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18 views

How to perform statistical test for two sets of points?

(I have asked this question originally on Cross Validated; however, no good answer and someone suggested me to ask the question here). Thanks a lot in advance if anyone can help. We know that we can ...
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34 views

Testing of hypothesis

Following is a question from my textbook. My approach is different from one explained in the book. I cannot understand what is wrong with my solution. I have explained both solutions below. Kindly ...
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25 views

Showing that $S_{xx} = \sum_{1}^{n} x_i^2 - \frac{(\sum_{1}^{n} x_i)^2}{n}$

I am having problems understanding the identity (more specifically the last equality) $$S_{xx} = \sum_{1}^{n}(x_i - \bar{x})^2 = \sum_{1}^{n} x_i^2 - \frac{(\sum_{1}^{n} x_i)^2}{n}. $$ I've made an ...
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Derivative of Riemann-Stieljes integral?

In pp. $5-6$ of Roger Koenker's Quantile Regression, the author minimizes the function $$(\tau-1)\int_{-\infty}^\hat{x}(x-\hat{x})dF(x)+\tau\int_\hat{x}^{-\infty}(x-\hat{x})dF(x)$$ with respect to ...
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16 views

Combined Effect size

Is there a way to calculate the effect size between more than 2 components? For example, if i know the effect size of variable A on C and I also know the effect size of variable B on C, is there a ...
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23 views

what is the unit of the following standardization formula?

If normalization would be to rescale the range of values to scale between $0$ and $1$ as follow: $x^{'}=\frac{x-\min(x)}{\max(x)-\min(x)}$ where the interval of $x^{'}$ is closed between $[0,1]$ ...
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12 views

Prior that incentives dissimilarity of 2 parameters

I have some binary data. I have a proposed partition of this data into partitions 1 and 2. I want to test whether the data in models 1 and 2 were generated by two Bernoullis such that their ...
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28 views

Find the mean and standard deviation of the sampling distribution of the restaurants sample mean expense per customer.

A restaurant charges $8.95$ pp. Management finds it's expenses per person has a distribution that is skewed to the right with a mean of $8.20$ and a standard deviation of $3.00$. Q: If $100$ ...
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24 views

The concatenation of two independent normal vectors is multivariate normal.

I've already read this question. By the definition I have, $$\mathbf{z} = \begin{bmatrix} z_1 & z_2 & \cdots & z_n \end{bmatrix}^{T}$$ is a multivariate standard normal vector if each ...
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44 views

Find the probability of messages reaching a plane via two separate antenna towers.

I am very confused on how to setup the below problem. Any advice is welcomed! A control tower at an airport has two antennas for sending radio signals to approaching planes. Each message is sent ...
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35 views

What is this matrix notation and how is it solved?

I've never taken a stats class, or linear algebra or much of anything that involves matrices. In one of my books they give me this as part of an example and it states, $$\binom{6}{4} = 15 \text{ ...
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38 views

Maximum likelihood estimator of $\lambda$ and verifying if the estimator is unbiased

$(X_1,...X_n)$ is a random sample extracted from an exponential law of parameter $\lambda$ Calculate the likelihood estimator $\nu$ of $\lambda$. Then, if $n=2$: establish if $\nu$ is a unbiased ...
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242 views
+200

The root of summation function

This is a calculation I need for my statistics project Big edit: simplify the function $f(x)$ a lot. Define for $f(x)$, $x\geq 0$, $$ f(x):=\sum_{k=1}^\infty ...
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21 views

Constant Moment Generating Function and degenerate Random variable .

Let $\{X_n\}$ be such that $X_n$ has a binomial distribution with parameters $n$ and $p=\lambda /n$ , then as known $X_n$ will converge in distribution to $Y$ which has a Poisson distribution with ...
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83 views

How to take the inverse of the matrix $X^{T}X$, when it isn't invertible?

If I have a matrix $X$ and I am trying to compute $(X^{T}X)^{-1}$, which is the inverse of $X^{T}X$. However, each time I try to do it in some computing package like R, I get that $X^{T}X$ is ...
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14 views

How to compute the chances of wining on this game?

In a game, you start with K coins. On each turn, a coin is flipped. If heads, you gain a coin; if tails, you lose one coin. Turns are played continuously until either you have 0 coins (you lost), or ...
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23 views

How to determine which of the 6 columns of this matrix are not linearly independent when combing with the rest?

I currently have a matrix $G$ with $6$ columns from a simulation that looks like: $$\begin{bmatrix}{} 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 & 0.0 ...
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Probability and Basketball

The professor asked us to imagine a scenario where we have a basketball player who isn't good at shooting free throws. He makes his first free throw with probability $0.2$ After the first free throw, ...
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Maximum Likelihood Estimation with 2 parameters for a Poisson distribution

I have two observations from a Poisson distribution. The first one ($N_r$) come with a Poisson distribution with mean $k_1$. For the second one ($N_e$) I know that $N_e - M$ also come from the same ...
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32 views

Can a statistical model have two unknown parameters?

Is an $N(\mu,\sigma^2)$ distribution a statistical model if both the parameters are unknown? The definition I have in front of me only refers to one unknown parameter.
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19 views

Variance in the sum of batch-correlated residuals in a regression

I am looking at a regression model of the following form: $Y=intercept+\beta_{Yf.n}X_f+\beta_{Yn.f}X_n +error$ where $X_f$ and $X_n$ are predictors. A value for $Y$ will be sampled from the ...
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42 views

What is sample variance of sample variance, and what is theoretical sampling distribution?

I am trying to work some things in R and I am having trouble understanding some of the instructions. I generated $1000$ samples of size $5$ from the standard normal distribution, and I calculated the ...
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19 views

What does this symbol mean for the MLE method?

This is an implementation of the maximum likelihood method on $\hat \pi$. I am unsure what that $\mathbf 1$ looking symbol means. MLE estimate of $\pi_y$ is ...
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Show that $\hat{\theta}$ is an unbiased estimator of $\theta$

Let $f(x, \theta) = \frac{1}{\theta} x^{\frac{1-\theta}{\theta}}$, where $0 < x < 1$ and $\theta > 0$. Let $X_1, \dots, X_n$ be iid with density $f$. Taking the log likelihood, I found $$ ...
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What are numerical methods of evaluating $P(1 < Z \leq 2)$ for standard normal Z? [closed]

Let $Z \sim Norm(0, 1)$ and denote its PDF and CDF by $\phi$ and $\Phi$ respectively. Then, theoretically, $P(1 < Z \leq 2) = \Phi(2) - \Phi(1).$ However $\Phi$ cannot be expressed in closed form, ...
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Does a larger coefficient of variation mean larger variability?

I'm reviewing right now for an exam and I've stumbled across this online reviewer. In the solution for question #2 it stated that Section B is more consistent so there is greater variability in the ...
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34 views

Using Maximum Likelihood Estimation

Im trying to verify the following: \begin{align} &\mu_{ML}=\frac1N\sum_{n=1}^Nx_n\\ &\sigma_{ML}=\left[\frac1N\sum_{n=1}^N(x_n-\mu_{ML})^2\right]^\frac12 \end{align} from: $$\ln ...
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17 views

t-distribution - sample size - degrees of freedom

I am reading about t-distributions in a rather non-rigorous book. An example is given here where the sample size is 25. The example also gives a numerical table used to determine the probability of ...
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34 views

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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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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39 views

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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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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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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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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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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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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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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Density Histogram interpretation in R

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