# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### find the maximum where gamma is attained

I have this problem and I am not figuring the beginning
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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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 $\lambda_2$, ...
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### 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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### 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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### 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. ...
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### 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....
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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: ...
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### 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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### 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 ...
### 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}=...