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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Identifiying Biased and Unbiased Samples

My little nephew asked me a question about biased/unbiased samples in which is teachers answer is something I disagree with to say the least (I don't agree with the assumption made by the teacher nor ...
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6 views

How to calculate a covariance matrix with given Canonical Correlation Analysis components and given variances/covariances for CCA components?

So given a covariance matrix, the Canonical Correlation Analysis (CCA) components can be computed along with the correlation between corresponding pairs of CCA components. What about the other way ...
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328 views

Poisson Distribution for Consecutive Figures

I am trying to find the probability for a Poisson distribution. The mean is two cars sold per day. The question is: "What is the probability that at least one car is sold for each of three ...
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3answers
39 views

Where does this conditional probability law come from?

I was trying to follow a computation done in my class notes, and was having difficulty seeing the inspiration for a part of the manipulation in a question regarding probability. I did some Googling, ...
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1answer
57 views

How to define the 'error'

I have true data $G$ and wrong data $F$. Both are $n$ dimension vector. $G\in \{G_i| 0<G_i<255\}, i = 1:n$. Because the ...
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1answer
29 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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0answers
17 views

Probability of Big Leads in CoinTossing Game Evaporating

In a Coin Tossing game, how is the probability of a lead change affected by the size of the lead that one side has - taking into account the number of coin flips remaining in the game? I have asked ...
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49 views

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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39 views

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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1answer
29 views

$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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10 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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22 views

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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15 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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1answer
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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1answer
26 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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2answers
51 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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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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1answer
352 views

Cramer von Mises test statistic

I am trying to derive the Cramer von Mises test statistic $$nC_{n}=\frac{1}{12n}+\sum_{i=1}^{n}\left(U_{(i)}-\frac{2i-1}{2n}\right)^2$$ where $U_{(i)}=F_{0}(X_{(i)})$ the order statistics from the ...
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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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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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3answers
23 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 $Y$? ...
4
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1answer
81 views

Prove that $\int_{0}^1 u^{\alpha_1-1} (1-u)^{\alpha_2-1} \, {\rm d}u =\frac{\Gamma(\alpha_1)\Gamma(\alpha_2)} {\Gamma(\alpha_1+\alpha_2)}$

I have the following equality in a textbook of mine $$\frac{y^{\alpha_1+\alpha_2-1} e^{-y/\beta}}{\Gamma(\alpha_1+\alpha_2) \beta^{\alpha_1+\alpha_2}} \cdot ...
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1answer
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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0answers
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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0answers
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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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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1answer
20 views

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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0answers
38 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
22 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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2answers
381 views

Get a Fisher information matrix for linear model with the normal distribution for measurement error?

For given linear model $y = x \beta + \epsilon$, where $\beta$ is a $p$-dimentional column vector, and $\epsilon$ is a measurement error that follows a normal distribution, a FIM is a $p \times p$ ...
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67 views

Control limit and population mean

A cola-dispensing machine is set to dispense $9.00$ ounces of cola per cup, with a standard deviation of $1.00$ ounces. The manufacturer of the machine would like to set the control limit in such a ...
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2answers
36 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
21 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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1answer
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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1answer
26 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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1answer
15 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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2answers
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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2answers
38 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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1answer
13 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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1answer
20 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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2answers
4k views

Calculating missing data points from standard deviation and mean

I have to figure out what to missing data points are from a set of 10 scores. The mean of the 10 observed scores is 20.0 and the standard deviation is 6.0. The observed scores are listed below, with ...
2
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1answer
23 views

Proof the concave transformation of the tail distribution is always above the tail distribution

I need to prove that for a given continuous non-decreasing distribution $F_X(x)$, and a concave non-decreasing distortion function $g(.)$ defined on $[0,1]$, the following holds: $$g(1-F_X(x)) \ge ...
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1answer
86 views

Understanding an application of Fubini's theorem

I'm going over some lecture notes for a course in statistical theory. There is a "proof" that the density for a $k$-dimensional multivariate normal random variable (with non-singular covariance) is, ...
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1answer
81 views

How Do I Find My Car

I have been discussing this problem with a coworker for a few days now and neither of us have made any headway on it. I would appreciate any help with a possible solution or maybe a suggestion of a ...
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1answer
26 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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1answer
27 views

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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19 views

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
185 views

Characteristic function

Question: Let $X_1$ and $X_2$ denote independent real-valued random variables with distribution functions $F_1$, $F_2$, and characteristic functions $\varphi_1$, $\varphi_2$, respectively. Let Y ...
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1answer
15 views

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 ...