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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Skewness of a difference of random variables?

In this article( http://www.diva-portal.org/smash/get/diva2:302313/FULLTEXT01.pdf )page 28 explains how to derive the skewness of a sum of random variables; I haven't been able to derive this ...
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1answer
9 views

Calculating decreased cost with increasing quantity

I have a hand made table I've been using to give customers price per unit on my items, which gives a better price for the more items that they buy. My sample table right now I need to keep the ...
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0answers
3 views

To sketch a “typical” plot of a specific time series model

Let X have a distribution with mean $\mu$ and variance $\sigma^2$, and let $Y_t = X$ for all t. Sketch a “typical” time plot of $Y_t$. My thoughts: This process $Y_t$ is stationary with mean $\mu$, ...
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11 views

marginal likelihood survivor function [on hold]

Suppose ($x_1$,…,$x_m$) and ($y_1$,…,$y_n$) are drawn from survivor function $S(x;e^{θ/2}) $ and $S(y;e^{-θ/2}) $ respecitively , where $S$ is a defined above. (i) The form of the marginal ...
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7 views

running an A/B test on skewed observations

I am having this question. I came with following explanation: If we run A/B test on right-skewed samples, we would be getting skewed results. I think If observations are having positive skewness ...
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7 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
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12 views

How to implement this formula

I'm trying to understand a formula so that I can implement it in C++. In SLAM, the robot moves and scans an unknown environment. In 2D laser sensor, a measurement $z$ to landmark (i.e. physical ...
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1answer
11 views

Statistics discrete random variable help? [on hold]

Hello I am having problems understanding the following linked text in my stats class. I was wanting if anyone could go over it for me and try to give me the main points for me to understand as I ...
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14 views

Expected size of set resulting from n random samples with replacement from population of size N [on hold]

If I am sampling n times with replacement from a population of size N, what is the expected size of my resulting sample set? How many distinct elements am I expected to get?
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1answer
23 views

Statistics Question (Probability) [on hold]

The Masterfoods company reports that bags of Peanut Butter M&M's contain 10% each brown and red candies, and 20% each yellow, blue and orange candies. The rest of the candies are green. Consider ...
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1answer
14 views

explanation of probability density function

How can we explain that if a random variable $X$ has pdf $f(x)$ then the function $Y=g(X)$ will have different pdf than $f(x)$ ?? And how to find the pdf of $Y=g(X)$ ??
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1answer
27 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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15 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
100 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
46 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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0answers
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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1answer
31 views

Indicator function property

The indicator function 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. \end{cases}$ Consider $N$ ...
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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
27 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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0answers
9 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
15 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
23 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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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
45 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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15 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
10 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
26 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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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
34 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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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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0answers
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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18 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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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 ...
3
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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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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 ...