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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drawing cards in the deck

Suppose 3 cards are drawn from a shuffled 52 card deck. The face cards are the Jacks, Queens, and Kings. Let A = {all diamonds} and B = {All face cards} Are the events A and B independent? ...
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Error in Six Sigma Rolled Throughput Yield Calculation

I've been looking at running some calculations on process yields and landed on Rolled Throughput Yield and Six Sigma. I had been using the product of all process yields in the past, no idea Six ...
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How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
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1answer
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finding UMVUE $P ( X_1> t)$

Suppose $ X_1,\ldots,X_n$ are i.i.d. random variables with density: $$f(x_i;\theta)=\theta x_i^{-2}$$ $$x_i>\theta$$$$\theta>0$$ The smallest order statistic $X_{(1)}$ is sufficient and ...
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How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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28 views

Confused with the power set of an integer

I am going through The Maximum Degree of a Random Graph by RIORDAN et al. On the second page, the notation $\mathbb{P}(\mathcal{D})$ is used which I assume the power set of the set $\mathcal{D}$. ...
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Deriving sample size using Hoeffding's Inequality

I want to use Hoeffding's Inequality to determine the necessary sample size $n$ to construct a confidence interval of $\epsilon$ and $\alpha$. I've consulted the Wikipedia article and am confused as ...
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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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1answer
36 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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28 views

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
13 views

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
35 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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1answer
31 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
48 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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1answer
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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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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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1answer
28 views

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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24 views

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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20 views

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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51 views

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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1answer
21 views

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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1answer
31 views

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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2answers
35 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
16 views

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
27 views

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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2answers
355 views

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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0answers
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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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1answer
38 views

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
18 views

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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1answer
37 views

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 ...