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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is the probability of having a specific pair of normally distributed variables depends only on the distance of their standard deviation from the mean

We have 2 independent random variables $a \sim N(0,1)$ and $b \sim N(0,1)$. I want a proof (or even good intuition) to show that the chance of having the pair $(a_0,b_0)$ is a function of only the ...
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27 views

Why not represent discrete multivariate probability distribution as univariate?

For example, a bivariate distribution over binary variables can be represented with a 2x2 matrix of probabilities: \begin{bmatrix} p_{0,0} &p_{0,1} \\p_{1,0} &p_{1,1} \end{bmatrix} The same ...
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1answer
26 views

Calculate average life time in system with inputs and observed items inside system in a given time

In a given time (t) i observe the inputs that enter a system and the total items that are the sistem, i.e ...
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10 views

Need help with hyper geometric distribution problem?

Consider a suitcase with 7 shirts and 3 pants. Suppose we draw 4 items without replacement from the suitcase. Let X be the total number of shirts we get. Compute $P(X ≤ 1)$ . This is a problem Marcel ...
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2answers
37 views

calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn

Let's assume we are talking about lottery that 6 distinct numbers from 1 to 40 are drawn. We know the lowest number that was drawn $X_1$. we have the random variable $X_6$ that represent the highest ...
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0answers
15 views

What is KL-Divergence? Why Do I need it? How do I use it?

I am currently studying KL Divergence. But It seems very confusing that I don't maybe understand why do I ever need it and what is that for? As I have been reading stuff about Mutual Information, it ...
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0answers
18 views

showing that a matrix has repetitive values?

Here my primary aim is to calculate the stationary distribution of a DTMC using left-eigen values i.e, $ \pi = \pi*P$. But for some matrices, I observe that some states a same stationary probability. ...
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0answers
19 views

Testing the population mean

Given a population size and a sample, how can i test if the population mean is above a certain value? Similarly, can i find out what what value the sample mean has to be to say that the population ...
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1answer
26 views

Calculating approximate growth from three numbers [closed]

I have a set of three numbers 3600, 5200,12000; how do I calculate an approximate 4th number ...
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0answers
15 views

Correct statistic test(s) for resting energy expenditure (REE) vs estimated weightloss percentage (EWL)

Hello I am a student wanting to do a statistics test to compare the following: REE (initial) vs %EWL after one year. REE (initial) vs weight loss after one year. Change in REE (initial to one year) ...
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21 views

Maximum likelihood in categorical distribution

Let $k$ be a known number of groups and let $X_1,\dots X_N$ be iid random variables each with density $f$ with regards to the counting measure. $$ f(x) = \sum_{i=1}^{k} 1_{\{i\}}(x) p_i $$ ($1_{\{i\...
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35 views

Expected value of $\psi(\sum X_i,\sum X_i^2) $

Let $X_i \sim ^{iid} N(\mu,\mu)$, and $\psi(\sum X_i,\sum X_i^2)=\frac{\sum X_i}{N}-\frac{\sum X_i^2-2(\sum X_i)^2/N+(\sum X_i/N)^2}{N-1}$ In a book I'm reading, it's stated that $E(\psi(\sum X_i,\...
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0answers
22 views

find the maximum where gamma is attained

I have this problem and I am not figuring the beginning
-1
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1answer
58 views

Probability of getting n heads when n is very large [closed]

How to show that the probability of getting n heads when 2n times coin is toss is very small? Moreover, how to show that the probability of more than n heads is close to 0.5?
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3answers
37 views

Probability using Bayes formula

Two masked robbers try to rob a crowded bank during the lunch hour but the teller presses a button that sets off an alarm and locks the front door. The robbers, realizing they are trapped, throw away ...
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1answer
10 views

Confidence interval for a density function parameter

I am trying to solve this problem: Given $X_1, \ldots, X_n$ a random sample of a population of random variables with p.d.f. $f(x, \theta) = e^{-(x-\theta)} I(x)_{x\geq\theta}$, find a confidence ...
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1answer
35 views

Use of language on wikipedia - what kind of distribution?

I have an interesting problem and was wondering whether anyone would be able to point me in the right direction. I am wondering whether the use of a word in the english language on Wikipedia is ...
0
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1answer
17 views

Estimating a parameter using the maximum likelihood-method and the method of moments

Let $X$ be a random variable that has a density function of the form $f_X(x) = (p + 1) x^p 1_{[0, 1]}(x), x \in \mathbb{R}$ where $p > 0$ is an unknown parameter. I now want to make an educated "...
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1answer
222 views

Puzzle question finding Calvin

How to solve this problem. I have reckoned that I need to take as optimization problem finding minimum value for waiting time. Any suggestions? Calvin has to cross several signals when he walks from ...
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23 views

poisson process question.

The following calculations arise in certain highly simplified models of learning processes. Let $X_1(t)$ and $X_2(t)$ be independent poisson processes having parameters $\lambda_1$ and $\lambda_2$, ...
3
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1answer
50 views

Should principles be proved? [closed]

I was wondering if one needed to prove principles. E.g., likelihood or condionality principles in Stats. Thank you!
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29 views

A “sloppy” proof of Neyman's factorization theorem

Could you please explain why the attached proof is called "sloppy"? What is wrong with it? Thank you!
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1answer
70 views

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common? I have got the total no of events by using 64 C 2. But I am unable to find the numerator(no. ...
2
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1answer
66 views

Can you suggest a method to generate random sample from following PDF?

‎Let‎ ${‎‎\bf{\alpha}}=(\alpha_1, \alpha_2, \ldots, \alpha_m)$ ‎and ‎‎$‎‎\textbf{b}=(b_1, b_2, \ldots, b_m, b_{m+1}).$ I intend ‎to ‎generate ‎sample ‎from PDF $$ g(\alpha_1, \alpha_2, \ldots, \...
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2answers
28 views

intuition behind having a unique regression line

I understand this mathematically. we have function of 2 variables represents the sum of square errors. We have to find the $a$ and $b$ that minimize the function. there is only one minimum point. But ...
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32 views

What is the computational complexity class of thermal photon statistics

I would like to know the computational complexity of the following formula for the variance of thermal photon statistics. $$P(n)=\sum_{d=1}^D\prod_{m=1}^M\frac{1}{(1+\langle n_m \rangle)(1+\langle ...
2
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4answers
84 views

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue? I am helpless regarding this. I don't know how to solve it....
2
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3answers
75 views

Why is $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$

I saw, for any continuous random variable $X$, $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$, where $a,b\in\mathbb{R}$, in my textbook. I don't quite understand why the openness/closeness of ...
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2answers
39 views

Find the Moment Generating Function $Y$. What is the distribution of $Y$?

Let $X_1$ and $X_2$ be independent normal variables with means 2 and 5 and variances 9 and 1. Let $Y = 3X_1 + 6X_2 - 8$. Find MGF. What is the distribution of $Y$. attempt: Im not sure about ...
2
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24 views

What is the relationship between the function $\mathbb{E}(Y \mid X = x)$ and linear regression?

Consider the function $$ r(x) = \mathbb{E}(Y \mid X = x) $$ This has been called the regression function in a textbook I'm using. I'm trying to figure out the relationship between this function ...
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2answers
33 views

How to generate a random variable $r_i$ such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$

How can I generate $r_i$ for $1 \leq i \leq n$, such that $\sum_{i=1}^n |\frac{r_i}{\sigma_i}|^2\leq\chi^2_{n,\alpha}$, where $\sigma_i^2$ is the variance of $r_i$ and, $\chi^2_{n,\alpha}$ is a chi-...
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2answers
74 views

Reference Request - Statistics Book with exercises

I'm looking for an as complete as possible statistics book with exercises, including the following topics: Probability Review Random Variables and Samples Descriptive Statistics Estimation (...
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0answers
35 views

Common Estimates Suggestions

Consider two Markov Chains $X-Y_1-Z_1$ and $X-Y_2-Z_2$ defined on same alphabet space $\mathcal{X}$, such that $Z_1= g_1(Y_1)$ and $Z_2=g_2(Y_2)$ for some functions $g_1,g_2$. Assume further that ...
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2answers
32 views

Looking for a good, rigorous book on Statistical Inference.

I'd like to brush up on my statistics, but most books are either overly "colourful" or just plain shallow, and certainly far from a Bourbaki-esque style of exposition. Is there any "Graduate Texts in ...
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1answer
29 views

Uniformly Choosing a number from a range [closed]

May you please help me how I can choose uniformly a number from a range. I have to use this for trust evaluation in social networks such as the following clause: Each user has a quality ...
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0answers
34 views

Lottery machine probability

Assume we have a lottery machine where you press a button and it returns one of 5 motifs in one of 5 colours. Each of these also has a chance to be gilded. Assume I have a dataset containing outcomes ...
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2answers
40 views

Derive the value of this probability analytically

Forgive me if this question is very basic but I genuinely tried to search around including this site and could not find anything that I could adapt to my understanding. ...
2
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1answer
17 views

Find the PMF for number of heads following the first tail on a four consecutive coin toss expriment

Suppose a fair coin is toss four times consecutively. Find the PMF for random variable of number of heads following the first tail. My take: Let random variable $X$ be the number of heads in this ...
1
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1answer
39 views

Relationship between averages of $x^tx$ and $xx^t$ for column vector $x$

If we have data set $x$ as $m$ of $n \times 1$ vectors, and we know the average over index $m$ of $xx^t$ is $<xx^t> = C$, where $C$ is $ n \times n$ matrix. What is the average of scalar $ x^...
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9 views

What should be expected values of weights in statistics (specifically a set of biweight weightings)

I am working on a project that involves taking the biweight sample variance of a velocity dataset. This is defined as $\sigma_{BI}^2 = N\dfrac{\sum_{|u_i|<1}(1-u_i^2)^4(v_i-\bar{v})}{D(D-1)}$ ...
1
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1answer
31 views

Simple Question on interpretation of Poisson dist. as approximation to Binomial dist.

I have a question regarding to the notion of using Poisson dist as an approximation to Binomial dist. I can easily prove $\lim_{n\rightarrow\infty} P(X=k) = P(Y=k)$, where $X \sim$ Binomial(n, p) and $...
3
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1answer
45 views

Variance of sum of linear combination

I want to calculate the variance of a sum of linear combinations, so $$\operatorname{Var}\left(w'R_1 + w'R_2\right)$$ where $w$ is a $N\times 1$ vector and both $R_1$ and $R_2$ are $N\times 1$ ...
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0answers
18 views

Problem in finding introductory material (matrix spectra)

I am looking for introductory material on: 1) matrix eigenvalue spectra and useful matrix algebra theorems that can be applied in the field. 2) Statistics of random matrices (i.e. ensembles, ...
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2answers
37 views

Probability question with a radio competition

I'm quite new to statistics and I'm going through a few exam questions but I am a bit stuck on this one: ...
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0answers
14 views

Derive the Hat Matrix to map actual response to estimated resposne

In order to measure the quality of a regression we can calculate the Hat Matrix. Using it we can estimate the response variable as if we used the predictor variables to regress them. For linear ...
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2answers
37 views

Probability of incorrectly spelling a word

I'm currently trying to teach myself Statistics and have an exam question that I need a bit of help on: ...
0
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1answer
14 views

loss function similar to normal density

let $$L_\epsilon(x,p) = -\frac{1}{\sqrt{\epsilon}}\exp\left\{-\frac{(x-p)^2}{\epsilon}\right\}$$ be a loss function. given a random variable $X$ with density $f$ (possibly restricted), the risk ...
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1answer
40 views

If I know the mean, median, mode, and so on how do I determine the standard PDF that most fits

I have been playing a game and it drops currency with a drop rate that has an expected value of $1/750$ and nothing else. I was able to keep track of 3000 drops and times between them. I can then find ...
2
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3answers
39 views

Tricky permutations question

There are 8 buckets, each bucket is a different color (for simplicity, let's label the colors A, B, C, D, E, F, G and H; if you like: Aqua, Brown, Cyan, Diamond, Eggshell, Fuchsia, Green, Hot-pink). ...
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0answers
27 views

MLE and unbiased estimator of $P\{X_{i}=1\}$ given poisson distribution

$\{X_{i}: 1\leq i \leq n\}$ is an i.i.d. Poisson random sample with unknown mean $\lambda$. Find the MLE of $P\{X_{i}=1\}$. Is the MLE unbiased? Does there exist an unbiased estimator of $P\{X_{i}=...