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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Combining Sample Means and Variances

Let us assume we have several samples of unknown size but known mean value $\mu_{i}(x)$ and known variance $\sigma_{i}^{2}$. Now we want to calculate the mean value and variance for the total ...
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10 views

Solving an integral to solve a statistical problem

In solving a statistical problem, I want to know $$\mathbb{P}( (Y_2| Y_1 \geq q_\alpha) \leq q_\alpha)$$ where $Y_1 = X_1 + Z$ and $Y_2 = X_2 + Z$ and $Z \sim N(0,\sigma_z^2), X_i \sim N(0, ...
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12 views

Does Weighted avg of Weighted avg makes sense..?

Scenario is like we have a city in which there are many schools. All of them conducted a mathematics exam and it depends on the school for each class how much marks the exam would be. Suppose school ...
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2answers
19 views

Symmetry in Probability Around a Particular Phenomenon in Time?

This has been hurting my brain substantially, recently. I'm not sure if I'm failing to make connections or if I see connections but am weary of their relevance. In my text the author claims that ...
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6 views

What is Bootstrapping in statistics? How can I use it to determine error in the mean, variance, kurtosis and skewness of a data set?

From what I understood from searching randomly is that it has something to do with resampling. What does this resampling mean? Is it selecting random data from a distribution or is it getting data ...
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11 views

Explaining the reasoning behind the initial derivation steps for the Gamma distribution

When deriving the gamma distribution, how is it justified that $P(W>w)$ is the summation of separate exponential distributions? I understand that for the gamma distribution, we have a series of ...
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1answer
11 views

How to prove that the maxium likelihood estimator of $\theta$ is aysmptotically unbiased and cosistent

In a class we looked at this example: Let $X_1,...,X_n\sim U(0,\theta)$. Then the maximum likelihood function is $\mathcal{L}(\theta) = \begin{cases} \dfrac{1}{\theta^{n}} & \text{if } ...
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1answer
12 views

Independence - Probability and Statistics

Any help on this problem is greatly appreciated! I'm completely stuck School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A ...
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1answer
478 views

How to quantify the differencen between 2/4 and 20/40?

Assume I have two methods to do prediction. The first method makes 4 predictions and 2 out of 4 are correct. The second method makes 40 predictions and 20 out of 40 are correct. The prediction ...
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1answer
31 views

If I flip $n$ coins, what is the probability of at least $3/4$ of them coming up heads?

I'm looking for an equation that will work for any positive integer n. I know that multiple coin flips approaches a normal distribution, but I'm looking for answers that deal specifically with random ...
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18 views

$\overline{X} \rightarrow c$ in probability $\Rightarrow g(\overline{X}) \rightarrow g(c)$ in probability

Prove that if $\overline{X} \rightarrow c$ in probability and if g is a continuous function, then $g(\overline{X}) \rightarrow g(c)$ in probability. Once I think in the situation $|g|<M$, it is ...
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1answer
22 views

Lagrange primal and duality properties

max Lprimal(x,lamb,mu) >= f(x*) + lamb g(x*) + mu h(x*) s.t. lamb >= 0 min Ldual(lamb,mu) = min max Lprimal(x,lamb,mu) What is the purpose and properties of the lagrange dual function, why do I need ...
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16 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i), \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
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18 views

Confidence interval of exponential random variables

I have a sequence of random variables $X_1, X_2, ..., X_n$ such that $X_i = e^{-(x_i-Θ)}$ I have to construct a confidence interval of the form $[Θ−c,Θ]$,where $Θ = \min _i{X_i}$. For $n = 10$ how ...
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1answer
67 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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2answers
13 views

Finding the probability of time between two events for a poisson process

If two events occur at a rate of 1.8 per hour on average, and this occurrence follows a poisson process, what is the probability that there is at least 1 hour between two events? My approach for this ...
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12 views

Stat MGF question [on hold]

Suppose we are given a random variable Y with probability density function f(x) = exp(((xƟ -b(Ɵ))/Φ) + c(y,Φ)) show that the moment generating function is: exp((b(Ɵ+Φt) -b(Ɵ))/Φ) Any ideas on how ...
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1answer
37 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
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0answers
35 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
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1answer
160 views

Why should statistics be considered mathematics? [on hold]

Statistics should be completely separate from mathematics because it is not exact as pure mathematics , so my question is why they are still together? Statistics is a science in my opinion, and it is ...
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1answer
23 views

Confusion about the sample mean and random variables

As I understand the sample mean you just add a bunch of random variables that constitute a sample from their common distribution and divide by the number of those same random variables. When I apply ...
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1answer
26 views

What does “2- place real function” mean?

What does "2-place real function" mean? This comes up in the context of copulas, as here.
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24 views

Estimating the size of my population

I have a following problem: Imagine you have a hat with many different balls in it and you want to estimate, how many balls are totally in the hat. The only think you are allowed to do is to take one ...
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172 views

Pareto distribution,fisher information, confidence interval [on hold]

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
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1answer
50 views

Density functions and estimators

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...
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1answer
30 views

Confidence Interval Question - Confused on Approaching the Problem

My question is almost exactly the same as another one here on math stack exchange, but it isn't as explanatory as I'd like it to be with some parts. I was unsure of whether or not it'd be "okay" to ...
2
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1answer
27 views

Probability of co-occurence

Of total $N$ people, $m$ people are good at mathematics and $c$ people are good at computer science. What is the expected number of people good at both mathematics and computer science? Or what is the ...
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1answer
27 views

proof of the convergence of confidence intervals

The confidence interval can be derived intuitively by replacing the standardized mean with the standard normal and variance with sample variance, but is there a formal limit? I'm trying to prove if ...
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1answer
64 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...
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12 views

finding min and max after removing percentage of num values knowing the standard deviation.

I have a question i have some data and i know it's (number of values, min, max, mean and standard deviation) can I know the minimum after removing x% of the total number of values and the maximum ...
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15 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
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1answer
28 views

Which statistics test to be used? [on hold]

I want to run a statistics test in SPSS that tests the potential differences in two sets of data. The first set contains the following ten values: 13.7, 34.0, -68.0, 19.8, 20.9, 23.1, 22.5, 18.4, ...
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1answer
46 views

Prooving a sequence of iid within an interval [on hold]

How would I solve part 3 given that I have done the previous questions?
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2answers
292 views

Average distance between two randomly chosen points in unit square (without calculus)

Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can ...
3
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1answer
41 views

How to prove the sign test

Please correct me if I'm wrong, but a version of the sign test assumes under $H_0$ that there is some distribution $F$ where $X_i \sim F, Y_i \sim F$ and $X_i, Y_i$ are iid. Then it states that $T = ...
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3answers
36 views

Probability of a Rare Event Occurring within a Certain Number of Times

I'd like to know how to find the probability of an event occurring, given the probability of that event, within a certain number of chances for it to occur. For example, say that once every year ...
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1answer
80 views

How to minimize the expectation?

Given random variables $X_0, X_1, \ldots, X_n$ with finite expectations $m_0, m_1, \ldots, m_n$ I want to prove that the numbers $a_i = \frac{\det \Lambda_{i0}}{{\det \Lambda_{00}}}$ minimise the ...
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1answer
21 views

statistics testing a claim please help. homework problem and stuck [on hold]

Systolic blood pressure levels for men have a variance of 388.09 mm^2 HG. a random sample of 31 women resulted in blood pressure with a variance of 497.29 mm^2Hg. Use a 0.05 significance level to test ...
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15 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
3
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1answer
14 views

Bivariate Normal Distribution Problem vs Marginals

If $X_1 \sim N(0,1)$ and $X_2 \sim N(X_1^2, 1)$ then does $(X_1, X_2)$ follow a bivariate normal distribution? My thinking is that $X_1 ^2$ is $\chi^2_1 $ since it's the square of a $N(0,1)$ random ...
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22 views

Combination of historical and current data in statistics

I have a general question about a statistical matter. Lets assume there exists a true and unique probability $p$ such that an event $X$ happens in the next 12 months. There is some information about ...
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32 views

Algebra Integral simplification

Let some equation problem final result is like this $0\leq1\leq s\leq t\leq u\leq v$ \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s ...
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1answer
29 views

Standard deviation: calculating how polarizing a question is

I'm trying to calculate how polarizing a question is. Let's say I have a question that has 3 possible choices. A certain percentage of people choose a specific answer. Answer a: $30\%$ Answer b: ...
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3answers
38 views

2 restaurants located randomly

any help on following question will be much appreciated. Q. Suppose that $2$ restaurants are going to be located at a street that is $10$ km long. The location of each restaurant is chosen randomly. ...
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1answer
91 views

Sum of i.i.d. random variables and finding an upper bound

Problem: Suppose that $(X_i)_{i\in\mathbb{N}^+}$ is a sequence of i.i.d. random variables. For some $n\in\mathbb{N}^+$, let $S_n=\sum_{i=1}^n X_i$. Furthermore, let $a$ be a positive constant, and ...
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1answer
28 views

Challenging probability and statistics problems?

The books I have on the subject lack entertaining problems. Can someone suggest a book with more challenging problems - perhaps not at olympiad level, but a bit easier. Thanks very much!
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25 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
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15 views

If the null hypothesis is true, how will the test statistic be distributed?

I went with T~(50-6) The question goes.... "A regression is estimated with 50 observations, five explanatory variables and with a constant. Suppose You want to test the following hypothesis $H_0: ...
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8 views

Confidence interval for mean using t-distribution (unknown pop.variance). Is the following statement true or false?

This is from an A-Level statistics paper. We are the given the lengths of 10 components and asked to calculate a 95% confidence interval for the mean. We are given that the lengths are normally ...
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3answers
45 views

Parameters estimation for gaussian function with offset

I've read the paper Least square fitting of a Gaussian function to a histogram by Leo Zhou on how to perform a Least Square Fitting of a gaussian function to a histogram. The Gaussian function used ...