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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Normal distribution where variance depends on mean

Let $x = \bar{x} + \epsilon$ where $\bar{x} \sim \mathcal{N}(\mu,\sigma^2)$ and $\epsilon \sim \mathcal{N}(0,\sigma_\epsilon^2(\bar{x}))$, i.e., the expected value of $x$ is normally distributed, plus ...
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10 views

Do we need to check that maximum likelihood estimator is a maximum?

For maximum likelihood estimation, do we theoretically need to check that the critical point is a maximum (rather than a minimum or saddle point) or is this automatic? I believe that it is automatic ...
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7 views

Procedure to determine unbiased and consistent estimator of moments

Preliminary definitions I have a random variable $X$ and $N$ independent observation of it ($X_i, i\in\{1, \ldots, N\}$). I know that: $$\mathbb{E}[X_i^r] = \hat{\mu}_r,~ \mathbb{E}[(X_i - ...
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11 views

Mean Preserving PDF Spreading

I have a univariate discrete random variable and a histogram representing its PDF (which is asymmetrical). Is there a known way to increase/decrease the variance of the distribution (i.e. scaling it ...
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1answer
34 views

Conditional Probability of A given B, is it not just A?

If Conditional Probably is defined as $P(A\mid B) = \frac{\displaystyle P(A \cap B)}{\displaystyle P(B)}$, and $P(A \cap B)$ is defined as $P(A) \times P(B)$, is $P(A \mid B) = P(A)$?
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27 views

The top 1% own 50% of the world's wealth - how do we turn this into a function?

This Oxfam report states that 1% of the world's richest own 50% of the wealth. But to be in the top 1% - you don't have to be a billionaire (assuming a billion is US dollar one thousand million). ...
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10 views

Reverse engineering a cumulative probability graph and determining the calculations necessary to create it?

I'm attempting to provide my own implementation of this graph, which shows how the probability of a scheduled rocket launch increases as the actual launch date grows closer - a low probability at a ...
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5 views

Showing that moment estimates are asymptotically bi-variate normal.

Let $X_1,\dots,X_n$ be iid $\Gamma(p,1/\lambda)$ with density $g_\theta (x) = \frac{1}{\Gamma(p)} \lambda^p x^{p-1} e^{-\lambda x}$, $x>0$, $\theta = (p,\lambda)$, $p > 0$, $\lambda > 0$. ...
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1answer
17 views

How do you explain the strong sampling assumption

When using the strong sampling assumption, we assume that our data points are drawn uniformly and independently. In the example I recently saw we have a data set: $D = \{16, 8, 2, 64\}$. And we have ...
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17 views

Show Projection minimizes variance

Van der Vaart's Asymptotic Statistics, problem 11.2 Another idea of projection is based on minimizing variance instead of second moment. Show that $\text{Var}[T-S]$ is minimized over a linear space ...
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9 views

check normality/accuracy of number based on a set

We have a set of numbers which collected by sensor, sensor produces a number based on some interval, and we store it on the set. Some time the sensor fails and produces incorrect number, how we can ...
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2answers
44 views

How to compute the sum of geometric distribution [on hold]

How to compute the sum of random variables of geometric distribution $X_{i}(i=0,1,2..n)$ is the independent random variables of geometric distribution, that is, $P(X_{i}=x)=p(1-p)^{x}$, then how to ...
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2answers
11 views

metrics for density-sampling similarity, beyond likelihood

I am looking for a metric that would evaluate the distance between a sample $S$ and a density function $D$ Building a sample from a known distribution can be done using a monte-carlo sampling, ...
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0answers
17 views

Exponential form of the Geometric Distribution

What is the exponential form of the geometric distribution? I am trying to understand how to determine whether a distribution is an exponential family by showing that the pmf (in this case) can be ...
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1answer
32 views

Expectation value of a product of n random variables

I am currently dealing with an expression of the form $\operatorname E[\Pi_{i=1}^n X_i]$, where $\operatorname E$ represents the expectation value and $X_i$ is an arbitrary random variable. ...
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18 views

Calculating variance and covariance of estimators. Where is the mistake?

I have a random variable $X$ and $N$ independent observation of it ($X_i, i\in\{1, \ldots, N\}$). We know that: $$\mathbb{E}[X_i^r] = \hat{\mu}_r,~ \mathbb{E}[(X_i - \hat{\mu}_1)^r] = \mu_r$$ I ...
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28 views

Existence and Uniqueness of an Estimator

The object to be observed consists of B cubes $(b_{1},\ldots,b_{B})$. The detector consists of $D$ parts namely $(d_{1},\ldots,d_{D})$. Let $p(b_{i},d_{j})$ denote the probability of detecting a ...
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7 views

F-test for nested models fitted over two curves with shared parameters

I am currently doing a numerical minimization routine to simultaneously fit two curves (with shared parameters) to two datasets. I've managed to show that, assuming the likelihood of the combined ...
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19 views

How to Distribute Points in a Poisson Distribution in a Circle

I have a circular area, and I need to distribute a certain number of points in this circle in a Poisson Distribution. Functionally, how would I be able to distribute the points in a Poisson ...
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1answer
15 views

When is the sample mean not efficient?

According to wikipedia, the sample mean is an efficient statistic for i.i.d. Gaussian random variables. Is there a class of one-dimensional distributions for which the sample mean is not efficient? ...
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16 views

Can you simulate from a cantor distribution?

Has someone run across a method for generating random variates from a Cantor Distribution? It seems like its abstract definition prevents this. In essence, can one "invert" the Cantor Function?
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9 views

Regarding SLR through origin via method of moments estimator

I am in big trouble. Consider the equation: $y_i = \beta x_i + u_i$ for i = 1, 2, ..., n The question requires me to estimate beta by method of moments. The result I acquired is: $ \beta = { ...
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28 views

Estimate probability density function of being in a certain time interval

​You arrive at a bus stop in an unfamiliar part of town. Assume that buses arrive at the stop with an unknown (to you) distribution and wait in the bus stop for a few ​minutes. The wait time ...
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1answer
35 views

How do I compare student pre-test scores with post-test scores to evaluate whether or not they “learned”?

We need to track student advancement in a topic based on pre and post test scores. That is, we give a pre-test on day 1 of class, then on the last day we give the exact same test, renamed as a ...
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3answers
33 views

Will someone explain this polynomial regression equation?

I am in high school and I need to write a program that does polynomial regression to any degree on a set of data for a personal project. I think that this Wikipedia Article has the equation that I ...
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8 views

Statiscal Distance Properties

Please anyone could give me any idea of how prove the following property of statistical distance: d(AB,CD)=d(A,C)+d(B,D) Remenber that: $(X,d)$---> Metric ...
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1answer
10 views

Relationsip between probability functions (density and cumulative)

If I have a CDF, I differentiate to find the PDF. But if I have a PDF, how can I determine the CDF? I don't know the constant that might've been there? For example, the exponential distribution has ...
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9 views

ANOVA model usability, variances are unequal.

I have the following data: It's about chickens laying eggs depending on what they are fed, and how they are fed it. I want to use ANOVA for analysing this, but the assumption of equality of ...
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2answers
42 views

How can we show that “almost surely” equal random variables have the same distribution?

How can we show that "almost surely" equal random variables have the same distribution? We know $X =\text{(a.s)} Y$. What I have so far: $$\begin{align*}\implies& P(X = Y) = 1 \\ ...
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1answer
27 views

Calculate probability of joint PDF

I'm given the following joint PDF and asked to calculate $P(X+Y>1)$ $f_X$$_Y$$(x,y)=2/5$ for $0<y<1$ & $0<x<5y$ and $f_X$$_Y$$(x,y)=$ $0$ else I know I have to take the ...
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8 views

Minitab Statistical Software

I am running a two sample proportion test in Minitab and I keep getting the P-value to be 0 on normal circumstances I know this can be legitamet answer. However, I also have run a one sample ...
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54 views

Using Jensen's inequality to prove the Cauchy distribution has no mean

I can see that there is no mean because $\int x / \pi(1+x^{2})$ does not converge from -inf to inf. But my prof hinted at using Jensen's inequality for the proof. $$f(E(X)) \le E(f(X))$$ How can I ...
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3 views

sum of standard deviation in hypotesis testing?

I am performing a test of difference between means with paired data. I have to compare the means of the time taken to complete a job that is divided in 3 different actions. I have the average time ...
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20 views

Find the probability of the event that number 5 and number 10 were chosen first and last, respectively at the experiment of choosing six numbers, if: [on hold]

a) the choices are with replacement and order is not count; b) the choices are without replacement and order is not count; The set is : {1,2,3,4,5,6,7,8,9,10} I am quite new at this so any help ...
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1answer
16 views

Building a compound probability distribution

I want to build a probability distribution for a "shock" variable. I want to show that there are p% chances of no shock, and (100-p)% chances of shock- in which case, shock is distributed according to ...
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1answer
28 views

Sum of residuals proof

Show that: $\sum x_i e_i=0$ and also show that $\sum\hat{y}_i e_i=0$. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved ...
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11 views

Method of moments estimator

Let $X_{1} ..., X_{n}$ be a sample from $U([0,\theta])$ for $\theta>0$ Find an estimator of $\theta$ by using the method of moment and next compute probability that $\theta_{0} < X_{n:n}$ for ...
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1answer
19 views

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$ i) Find the regions where the joint pdf of $(Z,W)$ is positive. ii) Find the ...
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60 views
+100

Expected Value of R squared

Let $n$ be a fixed positive integer. Generate $n$ numbers $x_1, x_2, ..., x_n$ from the set $[0,1]$, with the probability distribution being the uniform one and the $x_i$ all being independent of each ...
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1answer
20 views

Showing independence of two random variables

The problem is here The trouble im having is showing how $\bar{x}-\bar{y}$ is independent of $S_{pool}$. I know the covariance of ( $\bar{x}-\bar{y}$,$X_i-\bar{x}$)=0 and similarly for the other ...
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14 views

Variance of event counting

I have this question (not homework, review problem for qualifying exam), tried approaching it a couple of ways (unsuccessfully). Any recommendations? Let $X_1,..,X_n$ be i.i.d continuous rvs. A ...
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1answer
11 views

Simple multivariate probability

$$F(y_1,y_2)=1, 0 \le y_1 \le 1 , 0 \le y_2 \le 1$$ $$F(y_1,y_2)=0, elsewhere$$ Find $$P(y_1+y_2) \le \frac{5}{4}$$ I can solve this geometrically and I know the answer is $\frac{23}{32}$ I ...
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15 views

stochastic process involving cdf of a process [on hold]

I would like to know if anyone here could help me out with this exercise. Here it goes: A stochastic process is created from Yn = c(n)Xn, where Xn is a stochastic process with mean equals to zero, ...
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1answer
13 views

Distribution of linear combination of iid exponential rv

If $X_1,X_2,\dots,X_n$ are i.i.d exp$(\lambda)$. How can I find the distribution of $U_n = \sum^n_{i=1} X_i/i$? Is this CF, MGF, PGF related? Thanks.
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15 views

The distribution of the sum of a uniform random variable and a binomial random variable

I'm asked to find the distribution of $U=X+Z$, where $X\widetilde~R(0,1)$ - That is, $X$ has a uniform distribution for $x\in]0;1[$ $Z\widetilde~bin(1,1/2)$ - That is, $Z$ has a binomial ...
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1answer
29 views

How much more likely is X% than Y%? [on hold]

77% of group A like singing 52% of group B like singing How much more likely is group A to like singing than group B? And can I say it like 'twice as likely'?
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20 views

Getting Linear sum from sum of squares

I have sample space Ri = (44.19, 35.14, 57.23, 74.27, 47.27, 32.06, 21.75, 66.49, 25.01, 30.10, 77.53, 61.40, 74.45, 21.93, 20.84, 73.36, 82.62, 52.36, 40.23, 31.88, 1966.79, 2112.71, 1836.88), n = 23 ...
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9 views

Durbin-Watson test statistic

How would you go about proving the Durbin-Watson test statistic approximates to $2(1-r_1)$ where $r_1$ is the first sample autocorrelation function?
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1answer
17 views

Covering deficits with values with different weights

SO I have a couple of assessments with specific weights as follows: Assignment 1: 5% => Mark 60% Assignment 2: 5% => Mark 53% Assignment 3: 5% Assignment 4: 5% Test 1: 30% => 47% Test 2: 30% ...
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1answer
42 views

Choosing a Project Topic.

I am a undergraduate student and recently i have been assigned to project (one of courses). But i have to choose my own topic. I want to work in the field of ...