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

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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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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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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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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10 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
14 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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2answers
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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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13 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
16 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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8 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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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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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
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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
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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16 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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4answers
136 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
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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
33 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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1answer
28 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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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
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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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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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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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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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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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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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
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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
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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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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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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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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
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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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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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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 ...