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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Laplace approximation for binomial distribution in matlab

i using bionrnd() function to generate a random vector and Laplace approximation formula to approximate the binomial distribution. but Laplace histogram dose not ...
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Probability of woman receiving positive mammogram and having cancer

The probability that a randomly selected US woman will have breast cancer in their lifetime is 0.12. Women over 40 are advised to have regular mammograms because early detection of breast cancer means ...
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Expected valued of Random sums about dice and jar problem

A six-sided die is rolled , and the number N on the uppermost face is recorded. From a Jar containing 10 tag numbered 1,2,,,,10 , we then select N tags at random without replacement. Let X be the ...
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Conditional probability about card picking.

A card is picked at random from N cards labeled 1,2,3,,,,,N and the number that appears is X. A second card is picked at random from cards numbered 1,2,3,,,X and its number is Y. I am asked to ...
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1answer
3 views

Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
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1answer
21 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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12 views

Statistics on ordered lists

Participants of a study had to rank several proposition by their importance: 1st, 2nd, 3rd, and so on... Two cases are of interest for me: The "ideal" case: there are $n$ propositions and each ...
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1answer
16 views

Stuck on this probability question?

In a multiple choice test paper let us assume there are $m$ possible answers, if a person knows the correct answer to a question then they will mark the correct answer and that if they don't know it ...
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Finding better curved line of best fit

I have a set of hand generated data that follows somewhat closely to an exponential curve: I can come up with an exponential equation to the line that gives the values on the 3rd row, and Someone ...
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26 views

Simple probability exercise.

I have this question I need to answer but I'm a bit stuck having done no statistics before. I don't want any full solutions just hints please. A man can either drive or get the bus to his job. The ...
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1answer
23 views

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
61 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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1answer
15 views

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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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
30 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
13 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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30 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
27 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
19 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
12 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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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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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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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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18 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
18 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
14 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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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
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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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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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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
143 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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1answer
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, ...