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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Are the poll results likely to be valid if they are based on only those subjects who agree to respond?

I found this question in the book "Essentials of Statistics" written by Mario Triola. When Gallup and other polling organizations conduct polls, they typically contact subjects by telephone. In ...
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$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
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Find the radius of a cluster, given that its center is the average of the centers of two other clusters

I do not know if it is possible to find it, but I am using Kmeans clustering with Mahout, and I am stuck to the following. In my implementation, I create with two different threads the following ...
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14 views

Linear regression, analyze of correlation, remove a variable [on hold]

I have done a linear regression on a excel file in function of the variable A then I did an analyze of correlation. My teacher said that after that I can remove a variable but how to chose it ? My ...
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19 views

Comparing standard deviations of two unknown lists

Consider the following problem: List A contains 15 numbers in the range from 10 to 50. List B contains 15 numbers in the range from -50 to -10. Or any other two lists. Is there some trick to ...
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6 views

Develop relation between dependent and independent Using Tobit model

Depenent variable (Y): Range (0 to 10) (Not less than 0 and not more than 10) (range which i collected from field survey) Independent Variables: X1 - Time (in sec) X2 - Distance (in meter) X3 - ...
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Does really convergence in distribution or in law implies convergence in PMF or PDF?

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in ...
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24 views

$\mathbb E[\bar X_n]=0$

A conditional normal rv sequence, does the mean converges in probability, in this question how can i get $\mathbb E[\bar X_n]=0$? Here is my attempt; $$\mathbb E[\bar ...
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1answer
25 views

What does rotational invariance mean in statistics?

What does rotational invariance mean in statistics? The property that the normal distribution satisfies for independent normal distributed $X_i$, $\Sigma_i X_i$ is also normal with variance $\Sigma_i ...
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28 views

A conditional normal rv sequence, does the mean converges in probability

$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for ...
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49 views

What is my probability space and measurable space?

I have the following difference equation $$ \tilde{u}_k = \begin{cases} u_k & \text{if $\gamma_k = 1$, no signal lost} \\ \tilde{u}_{k-1} & \text{if $\gamma_k = 0$, signal lost} \end{cases} ...
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22 views

Range of a marginal density function?

Given the joint density function: $$p(x,y) = \frac{4x^3}{y^3} \text{ when } 0 < x < 1\text{ and } y > x; \text{ otherwise } p(x,y) = 0$$ How does one calculate the marginal density for $Y$? ...
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1answer
30 views

Statistics question involving exponential distribution and (maybe) gamma function

This is from a past stat exam that I am studying for my final tomorrow (lol). I believe this might have to do with gamma function. Could someone guide me through step by step of how to do this? An ...
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36 views

Intuitive explanation of “unit root” sought

I am struggling to get my head around the concept of "unit root" in relation to time series. And it would be a great help if someone could give me a two or three sentence intuitive explanation of the ...
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Suppose $40\%$ of the population possess a given characteristic … What is the probability $44\%$ or fewer possess the characteristic?

I have the following question: Suppose $40\%$ of the population possess a given characteristic. If a random sample of size $300$ is drawn from the population, then the probability that $44\%$ or ...
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1answer
36 views

Binomial distribution giving me an answer above 1?

I am doing the following question. If i have a box of $20$ soccer balls and the independent chance of a soccer ball of being flat is $0.1$. What is the probability of having at least $4$ flat soccer ...
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8 views

Distribution invariant to orthogonal transformations

Let $X,Y$ be two real valued stochastic variables and define $$ \begin{pmatrix} \tilde{X}\\ \tilde{Y} \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X\\ Y ...
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15 views

Grading system with multiple judges

This is a practical question, I'm not sure if it's on-topic here. So sorry if it's not. There is a competition where judges decide the score competitors by summing their performance mark in several ...
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2answers
35 views

Probability of getting a parking ticket

The city of Ithaca, New York, allows for two-hour parking in all downtown spaces. Methodical parking officials patrol the downtown area, passing the same point every two hours. When an official ...
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1answer
19 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
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37 views

Distinguishable balls in distinguishable boxes?

Suppose I have $n$ distinguishable balls and $N$ distinguishable boxes. A particular configuration of this 'system' is such that there are $k$ particles in a box, b, where $1\lt b \lt N$ (i.e. the ...
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1answer
14 views

Using chi-square test for statistics with multiple options in one variable

i think that this is more mathematical question, i am doing some statistics on survey. There is questions with multiple choices, so for example, if there are 4 choices, i can pick 1st ,2nd and 4th. ...
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1answer
25 views

Random Sample vs Simple Random Sample

I am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...
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13 views

Correlation Query

I can find $\mu_1, \mu_2, \operatorname{Var}(Y_1) \ \text{and } \operatorname{Var}(Y_2)$ but I am not sure how to get the co-variance of $Y_1$ and $Y_2$ in order to find the correlation of $Y_1$ ...
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41 views

Using the geometric distribution to find the probability that between 4 and 6 devices will be tested

Quality control tests spark plugs until they find one that doesn't work. If the probability of a spark plug working is 0.99, what is the probability that they will test between 4 and 6 (inclusive) ...
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21 views

Adding and Subtracting Normal Distributions

I think I know how to do this, but I'm not sure. I'm just hoping to check myself here before I do a bunch of work incorrectly. Suppose you have three independent normal distributions: Distribution A: ...
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1answer
19 views

Which function will fit this curve best?

I am trying to do a test of normality on this data set here. My QQ Plot looks like this . It looked like an arctan function to me. So my idea was to do a reverse "tan" function transformation on it. ...
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1answer
21 views

Analytic solution for matrix factorization using alternating least squares

The standard form for ridge regression aims to minimize the following cost function. $$ \min\ \ \sum_i(y_i-x_i^T\beta)^2 + \lambda\sum_j\beta^2_j $$ As described here, it's possible to differentiate ...
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1answer
12 views

Density function after a transformation.

Assume $Y$ has the density function $q(y) = 3y^2$ when $y \in (0,1)$, overwhise zero. Let $Z = -log(Y)$. Then; $$P(Z \le z) = P(-log(Y) \le z) = 1 - P(Y \le e^{-z}) = 1 - F_Y (e^{-z})$$ Then I ...
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How to do sampling for the following problem. [on hold]

There are 200 students with a mathematics exam marks. According to marks students are divided into five categories 0-20,20-40,40-60,... and I want to choose two random sample with 25 for a group. ...
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1answer
25 views

Assumptions of a probability distribution

Let $X$ be a continuous real-valued random variable indicating the fragility of a firm. Suppose that the firm defaults if $X$ takes a value above a threshold $u>0$. Hence $$ Prob(X>u) $$ is the ...
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Moment generating function of a uniform random variable.

My first attempt to this question was to find the first few moments about the mean and try to rearranging the those moments to obtain the general function as desired. However, when I tried to ...
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15 views

Finding variance with method of moments

In part (a), I know Var(Ni)=p(p-1) but how do I find the variance of the estimator with this result? And for part (b) of this question, I have to clue on how to tackle this question. I don't get ...
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1answer
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Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
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What distribution would describe this?

I start with 100 eggs, 10 of them being broken. I randomly select eggs without replacement until they are all split into baskets of 10 eggs each. Here's what I know: Best case scenario all 10 bad ...
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1answer
19 views

Statistics; Normal distribution question

I'm not sure if I am solving this question correctly A used-car dealership has found the length of time before a major repair is required on car it sells is normally distributed. Witha mean = 10 ...
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How do I find percentiles of data sets (Even vs odd)?

Given the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sample ...
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1answer
25 views

Does Binomial variables independence implies Bernoulli variables independence

$X$, $Y$ are independent variables with Binomial distribution. $X={\sum_{i=1}^nX_i}$, $Y={\sum_{i=1}^nY_i}$. $X_i$, ($1\le i\le n$) are independent Bernoulli variables. Same applies for $Y_i$ Is the ...
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1answer
36 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
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1answer
19 views

Finding method of moments

Question: In order to solve find the method of moments estimator, I know I need to first find the expected value of Y. But after finding $E(Y)= \frac{\theta}{2}$, what should I do next?
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Confidence level using unknown median

This is a general question. Lets say we are giving a sample space n. The sample space has an unknown median The order statistics of each person, $**X_{i}**$ in sample space is measured (lets say ...
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1answer
26 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
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26 views

standard deviation and adjusted R-squared for simultaneous regressions

I am conducting a study that requires two steps of statistical estimation. First, I run a regular OLS regression, from which I gather three outputs that I need: coefficient values standard ...
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27 views

Statistical research [on hold]

I'm preparing for an exam and I came across this question in a book. Below, I understand that we have to assumed a split plot in time analysis with 3 way factorial with 3 drug $\times$ 2 sex $\times$ ...
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46 views

Statistical research question [on hold]

I'm preparing for an exam and I came across this question in a book. An experiment is done to examine ways to detect phlebitis during the intravenous administration of a particular drug. Phlebitis ...
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1answer
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When using the Central Limit Theorem, how to scale the mean and variance depending on the number of samples?

So I'm reviewing my notes for the central limit theorem for my final and I'm getting hung up on one detail. The two questions below both utilize the central limit theorem, but they use it in ...
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26 views

case deletion formula restricted least square estimator [on hold]

hi any one please help me to find out a case deletion formula for restricted least square estimator? $$ \hat\beta = (X' X)^{-1} X'y-(X' X)^{-1} R' [R' (X' X)^{-1} R]^{-1} R(X' X)^{-1} X'y $$ i need a ...
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2answers
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How to show U1 and U2 are independent (Stats)

Prompt: Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are ...
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2answers
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explain how this confidence interval is correct [on hold]

An srs of $16$ households is selected in Houston and the number of remote controls is counted. we are interested in a $99\%$ confidence interval for the population mean number. In the sample, the mean ...
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Markov chain steady state existence

Is it possible for a Markov chain to have no steady state solution ? What is an example of such system ?