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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Urn with marbles, unknown number of colors

When I started with this calculation I thought this was going to be a flashback from school decades ago but now after searching I'm confused if I'm over thinking it or if it's not as trivial as I ...
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How to calculate probability of users generating distributed events reaching n events per 15 minutes?

We have games & apps that connect to services such as Facebook and Twitter to fetch information. These services have various rate-limit caps that you cannot exceed - typically based on a 15 minute ...
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An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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2answers
50 views

Central Limit Theorem, why $n \ge 30$?

This is what I think the technical statement of CLT is: If we consider $\overline{X}_{n}$ coming from a sample of $\mathcal n$ independent and identically distributed random variables $X_{i}$ with ...
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Unable to olve the MLE: difficulty in computing the derivative

The probability that there are $k$ observations within distance $t$ of $x$ can be written as : $$\mathbb{P}[N(t,k) = k] ={n-1\choose k}[f(x)H_t]^k[1-f(x)H_t]^{n-k-1}$$ The pdf of the distance from ...
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Performing a signed test on two samples

Here is the problem My book hardly has anything about performing a sign test so I am a bit lost. Basically I know that I am testing $H_a: \mu_{13}\leq\mu_6$ vs $H_0:\mu_{13}>\mu_6$ at a level of ...
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13 views

Proving $A_i=B_i$ (Set theory)

Suppose that $P$ is a probability on a field $F$. Consider three events $A_1,A_2,A_3 \in F$ so that $P(A_i \cap A_j) = 0$ for all $i ̸\ne j$. Let $B_1 = A_1, B_2 = A_2 \cap A^c_1$ and $B_3 = A_3 \cap ...
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1answer
25 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
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1answer
12 views

Does the statistical frequency of patterns manipulate the probability of a given event? [on hold]

This is a question I've encountered when I first read about the Gambler’s Fallacy, I'm really wondering why it's considered fallacious? Taking statistics into consideration, If you studied the results ...
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22 views

Odds of two specific Hold'Em hands and an exact board?

The goal is to solve the probability of a Texas Holdem hand occurring where at the river one player has quad $9$s and the other player has exactly a $9$-high straight flush. (For non poker players ...
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1answer
24 views

Help reading a scatterplot

In reading the following scatterplot Would it be correct to say there is a strong positive linear relationship between the two axis? If not, could you please explain why? Also is it correct to ...
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1answer
17 views

How do I find a confidence interval of x% given an mle of θ and a standard error of sθ? [on hold]

How do I find a confidence interval of $95$% given an mle of $θ = 0.0357$ and a standard error of $𝑠_θ = 0.0057$? This is from the following question:
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1answer
10 views

How to find a confidence interval given mle and standard error?

How do I find a confidence interval of $𝑥$% given an mle of $θ$ and a standard error of $𝑠_θ$?
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6 views

Concrete example : likelihood, comparing theory and practice (genetic linkage)

Crossing tomato plants should give birth to a new population of plants of 4 types that we'll call $P_1$, $P_2$, $P_3$ and $P_4$. According to theory, the ratios of appearance of these 4 should be ...
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4 views

What is asymptotic variance and how do I find it?

I don't understand the concept of asymptotic variance. Given the mle of a probability function, the likelihood function and the random variables how do I find asymptotic variance? What exactly is ...
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14 views

Does likelihood ratio has any meaning in this case?

Let $Y$ be a real-valued random variable and $X$ a random variable with realizations in $L^2[0,1]:=L^2([0,1],\lambda)$, where $\lambda$ is a Lebesgue measure. Consider some functional ...
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1answer
17 views

Why does a process only satisfy the Markov property if and only if the random times are exponentially distributed?

Given, for example, a birth death process with a set of jump times. These jump times have to be exponentially distributed in order for this process to satisfy the Markov property. Why is this? Why ...
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How do you calculate P(A/B), when event B occurred after event A?

There's really only one question I can't begin to handle when it comes to probability, literally. It's not the only type of question I struggle with, though it's the type of question where I can't ...
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16 views

Properties of expectation values

I have a complicated problem in which I have to compute the expectation in order to see if my estimator is unbiased or not. After a lot of calcutions at the end I have found: $$ ...
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1answer
17 views

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
11 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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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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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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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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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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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
12 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
15 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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28 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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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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4answers
140 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
51 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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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
34 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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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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29 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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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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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
26 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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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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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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11 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 ...