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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How do you derive the confidence interval of linear fit parameters?

I have read a few ways of deriving the expressions for the parameters of a linear fit (i.e. slope and intercept) for a given set of values $X$ and $Y$. However, I have not found a treatment of: The ...
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How to determine the correct $\epsilon$ value for comparison of real values of a custom data set?

When comparing two real values we usually define an $\epsilon$ value and say they are equal if $|a-b| < \epsilon$. I evaluate Pareto approximations using different performance metrics also known as ...
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28 views

How come this Poisson formula equals 1

In Poisson Random Variable: $$\sum_{x=1}^\infty \frac{e^{-\lambda}\lambda^{x-1}}{(x-1)!}=1$$ Why does this equal $1$? What property is this?
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Coupon Collectors Problem with Packets and Subsets

The Coupon Collector's Problem (CCP) is very useful in many applications. However, the "default" CCP is relatively simple: suppose you have a urn containing $n$ pairwise different balls. Now you want ...
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SQL joins and analysis

Say we have a users table and an events table and what sort of analysis can be done? Also, what is some SQL statements to describe the analysis of these 2 tables?
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How does one find the density of the $k$th ordered statistic?

Let $X_1,\ldots,X_n$ be $n$ iid random variables. Suppose they are arranged in increasing order $$X_{(1)}\leq\cdots\leq X_{(n)}$$ The first ordered statistic is always the minimum of the sample ...
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1answer
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Proving Weak Law of large numbers by Markov's inequality

Hi I am trying to solve the problem 5.13 of the book Statistical inference by George Casella and Roger L. Berger. The problem is Formulate and prove a version of the WLLN with a weaker assumption ...
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1answer
33 views

Probability of a random Permutation [on hold]

Pick up a random permutation in S5(assuming all elements have the equal chance to be picked). Find the probability that the sum of the first three entries of σ is less than or equal to sum of last ...
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27 views

Birth-death Process/Extinction

Random processes in Continuous time. Given that $\beta = \frac{4}{5}*\mu$ I have calculated that the birth rate $= 0.4$ and the death rate $= 0.5$. If the initial population $X(0)=6$, how many events ...
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How do you interpret conditional probability when two events are switched?

Before I pose my question, I want to emphasize that I am not seeking a homework help or steps on how to derive the answer, for I already know the solution, and how to get it. What I am seeking is, how ...
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1answer
46 views

In how many ways can $8$ appointments be scheduled for a physician visiting a nursing home with $20$ patients? [on hold]

A physician routinely visits a local nursing home on Thursday mornings to examine patients. Suppose the facility has $20$ residents, but the physician only has time to check $8$. The supervisor places ...
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Urn probability replacement problem

An urn contains $10$ red and $10$ white balls. They are taken out at random one at a time. Find the probability that the fourth white ball is the fourth, fifth, sixth or seventh ball drawn if the ...
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Derive the asymptotic distribution of $\frac{2}{n(n-1)}\sum\sum_{i<j}|X_{i}-X_{j}|$

Derive the asymptotic distribution of Gini's mean diference, which is defined as $\frac{2}{n(n-1)}\sum\sum_{i<j}|X_{i}-X_{j}|$. This is an exercise of Asyptotic Statistics by A.W. van der Vaart. I ...
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15 views

Finding percentile given distance between two percentiles.

The sales for a company are normally distributed with mean $\mu$ and variance $\sigma^2$. The difference between the $90$th and $40$th percentile is $500$. The $70$th percentile is $1700$. What is the ...
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Relation of OLS and GLS risks

Let $Y=Xb+e$, where $Y$, $X$ and $e$ are random (usual linear regression model). Does it hold with some high probability that $$(Y-Xb)^T(Y-Xb)< c (Y-Xb)^T X X^T (Y-Xb)$$
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mann's test for trend

To test the null hypothesis that a sample $X_{1},...,X_{n}$ is i.i.d. against the alternative hypothesis that the distributions of the $X_{i}$ are stochastically increasing in $i$. Mann suggested to ...
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How many degrees of freedom exist in an agglomerative hierarchical clustering?

The computational complexity of generating an agglomerative hierarchical clustering from n vectors is $O(n^2)$ (calculating the pairwise distance matrix) dendrogram example However, the total number ...
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1answer
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calculating the standard error of the mean?

The mean of a random sample of size $n = 35$ is going to be used to estimate the mean of a finite population of $N = 400$. Given that the population standard deviation is thought to be 9.355, what is ...
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Estimate ratio of two expectations by sample means

I have a question about the estimation of a ratio of two expectations. Suppose $X_{i}$ and $Y_{i}$ are two random variables with $i=1,\cdots,N$. We seek to estimate $\mathbb{E}X_{i}/\mathbb{E}Y_{i}$ ...
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Is this a special probability distribution?

Does the distribution function: $\frac{1}{\theta}e^\frac{-y}{\theta} $ Have a special name? If not, how can I find the variance? I keep running into a dead end when I try.
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Proving that a statistics is not sufficient (uniform case).

I am posting a similar question - in the previous one I put a wrong distribution, which changed the whole question. Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that ...
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statistic /normal distribution [on hold]

The exercise requires to determine the expected profit for one bottle of gass if the price is $40$ dollars for a bottle of gas , the cost of gas is $20[1+(x-100)]$ dollars/ kg of gas also the ...
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maximum likelihood estimators of a shifted gamma distribution?

i had this question in my exam but didn't know how to solve this apart from constructing the likelihood function and differentiating .but got stuck in the middle of nowhere.please help . the answer ...
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Why doesn't the approximate mean work for matrix having negative values?

Let $X$ and $W$ are $N \times N$ matrix where $x_{i,j} $ and $w_{i,j}$ are positive numbers. $for ( i =1;i <= N; i++)$ $ \qquad Mean_{x,i} = mean(X(i,:)) $ $\qquad for (j=1; j<=N; j++)$ $ ...
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expection of random variable when the index also follows som

i don't have any clue to this question but because here the index also foloows a certain distribution please help?
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How to put my knowledge of probability and statistics to practice

Background: I am a masters student in stochastic analysis. My course is very theoretical, which in general is fine by me, it is what I enjoy the most. From the more data-friendly subjects, I have (or ...
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Probability of picking yellow after red.

I have a bag with $8$ red apples, $4$ green apples, and $5$ yellow apples. I select two apples without replacement, what is the probability that the second apple is yellow if the first is red? ...
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Estimation of using method of maximum likelihood

PDF $f(x;\theta) = \frac{x}{\theta^2} \exp \left ( - \frac{x^2}{\theta^2} \right )$ obtain an estimator of $\theta$ using the maximum likelihood method i think the likelihood function would be ...
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Find $c=c(n)$ so $T = c \sum_{i=1}^{n} |X_{i}|$ is an unbiased estimator.

I'm having some trouble trying to solve the following problem: Assuming that $X =(X_{1},\ldots,X_{n})$ is a random sample from the normal distribution with mean $0$ and unknown standard deviation ...
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1answer
30 views

IMPROVED - Proving that a statistics is not sufficient (Gaussian case).

Let $X=(X_1,...,X_n)$ be i.i.d. $N(0,\sigma^2)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
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Why is the mean of the minimum of $100$ exponentially distributed random variables equal to $\beta$ divided by $n$?

Here's a question about order statistics, I can't seem to understand. Suppose a battery lasts $1,000$ hours. If I have $100$ batteries, why is it that the mean that the first battery will go out ...
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34 views

Does the sum of Poisson random variables have a Poisson distribution?

So I have been taught that the sum of Poisson random variables have a passion distribution. However, I have a problem with this. Suppose you have a Poisson random variable $X$ with $E(X) = a$. Then ...
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1answer
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Maximum likelihood estimator for a Poisson random variable given that the parameter is discrete.

Let $x_1 = x_2=x_3 = 1, x_4 = x_5 = x_6 = 2$ be a random sample from a Poisson random variable with mean $\theta$, where $\theta\in \{1,2\}$. Then, the maximum likelihood estimator of $\theta$ is ...
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1answer
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Use Maximum Likelihood Estimation to guess which dice got selected

We have two six-sided dice (faces numbered 1 through 6) and two tetrahedral dice (faces numbered 1 through 4). Someone selects two of them and throws each once. Then they tell us the sum of the ...
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Percentage greater than 2 standard deviations from the mean

A question reads: "The weights of $910$ young deer tagged and weighed in a research study are normally distributed with a mean of $86$ pounds and a standard deviation of $2.5$ pounds." Approximately ...
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Maximum likelihood estimate lies outside the paramater space

Say if I have a model where I impose the restriction that $\hat{\theta} \in (0,1)$, and I calculate the MLE to $\not\in (0,1)$, does this mean my model is incorrect, for this parameter restriction?
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When is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible?

The question is quite simple: for a $N \times p$ matrix $\mathbf{X}$ with real entries, when is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible (where $\mathbf{I}$ is the $p \times p$ identity ...
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1answer
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Given probability distribution $f(x)=2-bx$ find $b$ and range for $x$

Suppose that the distances between houses and the center of a city are distributed with the density function: $f(x)=2-bx$, where $x$ denotes distance. If this is a proper density function, what can we ...
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1answer
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Confusion regarding the weak law of large numbers

I can intuitively understand that as I take more samples from a random variable $X$ (gaussian distribution), the mean would approach $E(X)$. But what I don't get is if I look at it mathematically. ...
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3answers
55 views

What is the difference between 10% and $\frac{1}{10}$

In a national competition , ech school had to choose 10% of students who participated in the competition . So my question is , why they didn't asked for $\frac{1}{10}$ of students who participated ? ...
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Let $E(X)=\mu$ and $\operatorname{Var}(X)=\sigma^2$. If $E(Y|X)=a+bX$, find $E(XY)$ as a function of $\mu$ and $\sigma$.

I can't figure out the answer for a question on my econometrics course. Somehow it seems simple, but still I can't seem to figure it out. Maybe I am thinking the wrong way about it. Could someone ...
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1answer
19 views

Is Pearson's chi squared test the right method?

I have a sample of n=1000. The sample covers cars being brought in for service after one year of ownership in my country. For each car, I know which defects it had when it was brought in. I'm trying ...
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1answer
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Why is $E(X_2|X_1) = X_1$?

From textbook: $f(x_1, x_2) = 2 e^{-x_2/x_1},$ where $ 0 < x_1 < 1$, and $ x_2 > 0.$ The marginal is $f(x_1) = 2x_1$, and accordingly $$f(x_2|x_1) = \frac{1}{x_1}e^{-x_2/x_1}.$$ My ...
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limiting value of an expectaion of a sum of random variables [on hold]

Let $X_1,X_2,X_3,\dotsc$ be a sequence of i.i.d. $N(\mu, 1)$ random variables. Then, $$\lim_{n\to\infty}\frac{\sqrt \pi}{2n}\sum_{i=1}^n E\left(|X_i-\mu|\right)$$ is equal to ____________. ...
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Suggest an unbiased estimator for θ and provide an estimate for the standard error of your estimator.

If $Y_1, Y_2, \ldots , Y_n$ denote a random sample from an exponential distribution with mean $θ$, then $E(Y_i)=θ$ and $V(Y_i)=θ^2$. Thus, $E(\bar Y)=θ$ and $V(\bar Y)=θ^2/n$, or $σ_Y=θ/\sqrt{n}$. ...
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show that $Y_1$ is unbiased for $\theta$ and find its variance [on hold]

Let $X_1,\ldots,X_n \stackrel {\text{iid}} {\sim} \text{$P_0$}(θ)$ $$Y_1= \frac {X_1+3X_2+5X_5} {9} $$ $$ Y_2= \sum_{i} X_i$$ Show that $Y_1$ is unbiased for $\theta$ and find its variance. Show ...
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1answer
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Why does $E(C\cdot \epsilon\; \vert\; C\cdot X) = E(C\cdot \epsilon\; \vert\; X)$?

Let $C$ be an $n\times n$ matrix, $X$ is $n \times k$, $\epsilon$ is $n \times 1$ This is taken from a simply proof of strict exogeneity in an Econometrics textbook by Hayashi. The explanation he ...
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What is the probability that a psychic correctly “predicts” the outcome of at least 5 out of 10 coin flips?

Assume the psychic is actually just randomly guessing on each flip. The attempt: let E be the event in question number of outcomes per flip = 2 chance of correctly guessing the correct outcome = ...
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1answer
49 views

Troubles With The Beginning

The following is the question I'm having a bit of troubles starting: Musicnotes.com sells sheet music in the following genres: rock jazz, new age, and country. An experiment consists of recording the ...
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Finding joint distribution for the following

I am trying to do the following problem Suppose that $X_1,...,X_n\stackrel{iid}\sim N(0,1)$. Define $$\bar{X}_k=\frac{1}{k-1}\sum_{i=1}^{k-1}X_i,\,\,\,\,\,\,\text{for }k=2,3,.....,n $$ (i) What is ...