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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Probability of a random Permutation

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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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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33 views

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

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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14 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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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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17 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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14 views

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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28 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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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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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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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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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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351 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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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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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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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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54 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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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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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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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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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 ...
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Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify ...
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Is a random variate a functional?

A https://en.wikipedia.org/wiki/Random_variate is a particular outcome of a random variable. I was wondering what is meant in the Wikipedia article with "a random variate is corresponding to a ...
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Statistics: Attributable Risk.

$AR$ = Attributable Risk Suppose that we were determining the effects of smoking on heart disease. We found that among smokers in a certain age range, a proportion of $.4532$ with heart disease and ...
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Method for determining distributions of sum of Normal distribution unknown mean and variance

I've been trying to complete this question but have been struggling to see how to approach it. Any help would be greatly appreciated. Is there a standard way of approaching and answering ...
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50 views

Methods for calculating the mean and variance of a distribution created from the addition of two normally distributed quantities

I'm trying to understand how to interpret the following which refers to determination of the mean and variance of a distribution that's the result of adding two normally distributed random variables. ...
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Calculate $P[A,B,C]$ from $P[A,B]$ and $P[B,C]$

I have 3 (not independent) events $A, B, C$ and I know everything about how any two of them correlate. For example, I know: $$ P[A], P[B], P[C], P[A,B], P[A,C], P[B,C], P[A|B], P[A|C], P[B|C], ...
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Find probabilty

I have this table of information: Probabilities: \begin{array}{c|c} .919 & ????\\\hline ???? & .274 \end{array} How do I find the probabilities of the question marks? I thought each row ...