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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18 views

Determining the next observation with a 95% confidence.

Suppose $X$ follows a Poisson distribution with an unknown parameter $\mu$. The outcome of an experiment gave a value $X=625$. I want to determine, given this outcome, the interval in which the next ...
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11 views

Which significance test can be used in this instance? - paired, non-normal means [on hold]

I'm currently writing my final year dissertation for a Biology degree, and I am really struggling with the stats behind my results. I have a data set of behaviours for a herd of captive elephants ...
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3answers
85 views

Anyone can integrate $e^{-\frac{x^2}{3}}$ by hands?

I just used wolfram integral calculator and the result is weird, there is something called error function. $$ \int_{-\infty}^\infty e^{-\frac{x^2}{3}}\,\mathrm dx $$ Hint says that change of variable ...
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2answers
33 views

Tossing two dice with sum equal to 4?

Exercise: Throw two dice. Suppose that eye sum are 4. Calculate the resulting conditional probability that a) the first dice gave a 3 . b ) the second dice gave two or fewer eyes. c ) ...
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26 views

trying to understand binomial distrubition

I'm trying to understand when I can use the binomial distribution. I have searched some examples online and I'm wondering if I can use them in this situation: if we had a deck of 20 cards and we ...
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18 views

Indicator function property

The indicator function (random variable) for a probability event $A \subset \Omega$ is given by $ \mathbf{1}_A(x) =\begin{cases} 1 & \text{if }x \in A \\ 0 & \text{if }x \notin A. ...
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13 views

How would you run an A/B test if the observations are extremely right-skewed? [on hold]

How would you run an A/B test if the observations are extremely right-skewed?
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13 views

Find UMVUE for $X_1 \dots X_n \sim N(\theta, 1)$ [on hold]

Let $X_1, \dots, X_n \sim N(\theta, 1)$ with $\theta$ being the parameter we are trying to find. The question I am working on says as a hint to proving that $\bar{X}$ is the UMVUE, we should first ...
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1answer
24 views

Summation of binomial number of poisson random variables

Z is summation of K random variables that each has Poisson distribution with different means. But, K is a Binomial random with parameters of n and p. I was wondering what is the distribution of Z?
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6 views

Proof about an Inhomogeneous Poisson Process

We know that an inhomogeneous Poisson process is a process with a rate function $\lambda(t)$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
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0answers
13 views

How to find the CRLB: Unable to follow th steps in paper

I am unable to follow the steps needed to derive the Fisher Information matrix and the CRLB of an autoregressive model from the observations $x$. The AR process is excited by non-Gaussain sequence, ...
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1answer
18 views

Distribution of random variables when combined

I need help with this problem: If $X$ and $Y$ are two independent random variables and are both standard normal, what is the distribution of $\frac{1}{2}(X^2+Y^2)$? I think I start with ...
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0answers
4 views

A specific question on relative dispersion.

I'm trying to work out a problem involving relative dispersion, I think I have the right answer but something about it doesn't make sense. Here are the details: The Question: Consider a gas of $N_0$ ...
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1answer
25 views

Probability that sum of two uniformly distributed random variables is less than some constant

I am trying to find a way of determining the probability that the sum of two continuously uniformly distributed random variables is less than some constant $C$, formally: Let $A \sim ...
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0answers
5 views

To calculate type I error of hypothesis testing on a discrete random variable

Suppose X is a random variable with $P(X=k)=(1-p)^kp$ for $k\in{0,1,2,...}$ and some $p\in(0,1)$. For the hypothesis testing problem $H_0:p=1/2$ and $H_1:p\neq 1/2$. Consider the test "Reject $H_0$ ...
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1answer
39 views

What books do you recommend on mathematics behind cryptography?

I am currently reading the Book Understanding Cryptography from Cristof Paar. I am enjoying the book but i don't like to scratch the surface when it comes to cryptography. I would like do dig a little ...
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0answers
5 views

Experimental Pre and Postdesign, ANOVA

I have the following experimental design and would like to know how to apply ANOVA to the data: The experimental set up is as follows: Pretest, Treatment (or Training) and Posttest In each Pretest ...
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0answers
14 views

A math proof within a question about homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
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1answer
30 views

Binomial distrobution, find number of trials such that correct outcome occurs 99% of the time

An algorithm gives a correct answer with prob p=0.75. The output is binary (0 or 1). How many times should this be run with the same input such that the correct output occurs with probability at least ...
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0answers
9 views

how to find relationships in dataset with multiple variables

I have a large project data set ,which includes numeric values like dollar amounts, and non numeric quantities like country codes, purpose codes etc I want to find relationships between the variables. ...
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1answer
25 views

probability of two successive random numbers has the same starting number

Question/problem(subtask b): What is the probability of two successive random numbers has the same starting number? What we do know is that a random number generator randomizes numbers of 6-digits ...
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0answers
16 views

Does martingale model work for betting football matches?

Imagine I have 1 million USD and will be betting 1.000 USD on the win of FC Barcelona each time they play a match in La Liga (Spanish Tier 1 football league). If FC Barcelona loses or ties their last ...
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1answer
33 views

CDF of minimum of correlated and iid random variables

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are ...
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0answers
21 views

Joint probabilty density function

$f_{X,Y} (x,y) = x e^{-x(1+y)}$ if $x \ge 0$ and $y\ge 0$, $0$ otherwise. Find $f_Y(y)$ I started with $\int_0^\infty x e^{-x(1+y)} dx $, but I cannot come up with a right answer which is ...
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24 views

Finding the probability density function of $U=Y_1+Y_2$

Let $(Y_1,Y_2)$ denote a random sample of $n=2$ from the uniform distribution function on (0,1). (1) Find the probability density function $U=Y_1+Y_2$ (2) Find $E(U)$ I am unsure of how to bound ...
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1answer
12 views

Why is the mean for the F distribution not 1

A random variable $X$ has an F distribution. It has $p$ and $q$ degrees of freedom. I understand that $E(X)$ can be proven to be $\frac{q}{q-2}$ by integrating $xf(x)$. Why does the method below give ...
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2answers
19 views

Weighted Average Proof

Been stuck on this for a while now, seems pretty straightforward but can't seem to prove it. Given $\mu$ is a weighted average of $\mu_1$ and $\mu_2$ such that $\mu = x_1\mu_1 + x_2\mu_2$ where $x_1$ ...
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1answer
32 views

help with sample size calculation

I am conducting a study on a certain ratio in a blood exam (lets call it X) . I aim to say that there is a certain sensitivity (>80%) for people to be sick if $X>10$. its a retrospective cohort ...
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17 views

Math Economics question [on hold]

A consumer spends time $t$ searching for a good, the price of which is $p(t)$. Assume the longer the search goes, the lower price the consumer would pay for the good. Furthermore, assume there are ...
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0answers
31 views

Normal distribution where variance depends on mean

Let $x = \bar{x} + \epsilon$ where $\bar{x} \sim \mathcal{N}(\mu,\sigma^2)$ and $\epsilon \sim \mathcal{N}(0,\sigma_\epsilon^2(\bar{x}))$ are independent, i.e., the expected value of $x$ is normally ...
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1answer
44 views

How was the explicit closed form for this implicit function derived?

The problem comes from reading this [0] paper but I think I can express it in a self contained question. Consider the implicit function $H(z)$ defined by the relation: ...
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1answer
13 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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0answers
8 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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0answers
23 views

Mean Preserving PDF Spreading

I have a histogram representing the PDF of an unknown discrete RV. The histogram is asymmetrical. To be clear, all I have is the histogram. Is there a known way to increase/decrease the variance of ...
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1answer
35 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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32 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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11 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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9 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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23 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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10 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
53 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
13 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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21 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
35 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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0answers
20 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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0answers
30 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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0answers
11 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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0answers
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
16 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? ...