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Questions tagged [statistics]

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.

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Proving t distribution in small sample size and population variance known case.

I am asking a follow up question to this question. Why prefer the t-score when the sample size is low? I have seen mathematical proofs that in variance unknown case, the t-statistic follows a t-...
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2answers
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Expectation of the minimum of n independent Exponential Random Variables

Let $X_1$, $X_2$, · · · , $X_n$ be n independent Exponential random variables with mean 1. Find an expression for E(min($X_1$, $X_2$, · · · , $X_n$ )). I have been studying random variables for a ...
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Evaluate $P( \mid X+Y\mid \leq 2 \mid X \mid )$.

Let $X,Y$ independent random variables, identically distributed and symmetric about $0$. Select some probability density function f(.), and evaluate $P( \mid X+Y\mid \leq 2 \mid X \mid )$.
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1answer
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Log-likelihood function of the noncentral t-distribution

I'm trying to find an analytical expression of the log-likelihood function of the noncentral t-distribution as described here: https://en.wikipedia.org/wiki/Noncentral_t-distribution. It is ...
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2answers
26 views

Let $X,Y$ be independent, symmetrical, identically distributed, then $P( \mid X+Y\mid \leq 2 \mid X \mid )>1/2$.

Let $X,Y$ independent random variables, identically distributed and symmetric about $0$. Prove that: $P( \mid X+Y\mid \leq 2 \mid X \mid )>1/2$
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Given this transition probability matrix how do i go about computing the limiting probabilities?

[Question][1] [1]: https://i.stack.imgur.com/nV9Sa.jpg I was working on a problem to practice for an upcoming test and got stuck filling up the entries of the ...
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8 views

Expected value on order statistics, normal distribution

So I have this problem: Let $X_{(1)}, X_{(2)}$ be the order statistics of a random sample of size 2 of an N distribution $(μ,σ^2)$. Calculate $E(X_{(1)})$ and $E(X_{(2)})$.
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What is the meaning of the first and second derivative in order to find the Maximum Likelihood Estimator?

When trying to find the maximum likelihood estimator for a given parameter we usually proceed like this. 1.First we identify the distribution (Normal, exponential, Bernoulli, etc...) 2.We write this ...
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Calculating the Kullback Lieber information criterion in an application.

Let the observed random variables $X_1,X_2$ be independent and normally distributed with zero mean and variance respectively $\sigma_1^2$ and $\theta \sigma_1^2$, where $\theta \in \mathbb{R}$. The ...
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2answers
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How to find the last step required for finding the Maximum Likelihood Estimator?

We have a bernoulli distribution and we are trying to find an estimator for $p$. Here is the bernoulli formula $p\left( x_{i}\right) =p^{x}\left( 1-p\right) ^{1-x},x=0,1$ The joint probability mass ...
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2answers
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Assigning a value to how 'ordered' a vector is

Consider a vector which goes in ascending order from (1, 2, 3 ... N). This vector is ordered 'correctly' and I would like to assign it a score of 1, which indicates a perfectly ascending ordered ...
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Reconstruct a distribution using summary statistics and percentiles

I have a summary statistics and various percentiles generated from a data set. This include: mean, std, min, p10, p25, p50, p75, p90, p95, max Is there a way to reconstruct the data distribution from ...
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17 views

Prove these distributions are equivlanent

If we have that $X(t) = W(t)^2$ where $W(t)$ is the Wiener process. How do I solve the following problem? $ \begin{array}{l}{\text { Show that: }} \\ {\operatorname{Pr}\left\{X\left(t_{n}\right) \...
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22 views

Find the UMVUE for $\mu ^{2}$ by assuming $\sigma ^{2}$ is unknown.

Suppose that $X_1,X_2,...,X_n$ is a random sample from normal$(\mu, σ^2)$. Find the UMVUE for $\mu ^{2}$ by assuming $\sigma ^{2}$ is unknown. My approach: The distribution of the sample mean, ...
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27 views

How to calculate the density function of Z = XY where X and Y are independent random variables with uniform distribution in (0, 1)?

I need to know how to calculate which is the density function of a random variable Z that arises from the multiplication of two independent uniform random variables X and Y in the interval (0, 1) (Z = ...
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Crossover trials standard deviation/error

Hello, would anyone be able to explain how to get the standard deviation/error using this data? It’s supposedly 0.843 but I’m unsure as to how to get it. Many thanks in advance
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1answer
41 views

Conditional expectation of the number of phone calls

There are twenty individuals numbered $1,2,...,20$.Each individual chooses 10 others from this group in a random fashion,independently of the choices of the others, and makes one phone call to each of ...
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1answer
25 views

Edexcel A Level Statistics and Mechanics Year 1/AS Ex. 2E p.33 Q8 mean/standard deviation

I agree with the answer to (a). The problem I have is that if the answer to (c) is correct, then the mean would be (4+17)/2 = 10.5kn, which is different from the value 8.1kn from part (a), so ...
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0answers
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How to turn a non-stationary AR process into a stationary process? (ARMA modelling)

In my notes it says for non-stationary processes, forecast differences. E.G. For, y(t)= 1.5y(t-1)-0.5y(t-2)+ε(t) *: which is non-stationary as the coefficients sum to 1 Forecast, Δy(t) = 0.5*Δy(t-...
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Computational Statistics: Variance reduction

In this exercise, I should do the following subquestions for $\theta=P(X+Y\geq 15)$ where $X\sim Exp(0.5)$ and $Y\sim Exp(0.25)$ Implement the following estimators for $\theta$: (i) The raw ...
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Given two unbiased estimators, determine the optimal k* s.t. the corresponding θk* is the most efficient

Consider θa and θb are uncorrelated, unbiased estimators of θ and let θa(k)+θb(1−k)=θk. Determine the optimal k* s.t. the corresponding θk* is the most efficient (suppose MSE(θa) and MSE(θb) are σ^2a ...
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population standard versus sample size

I was wondering what is the difference between the terms of "population standard of 1%" and "sample size of 1%" for the assessment of uniformity. Thanks
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3answers
39 views

Find distribution of $Z=\frac{X+Y}{2}$ given $f_{X,Y}(x,y)=e^{-(x+y)}$

Excercise Let $X, Y$ be random variables such that their joint density function is defined by: $f_{X,Y}(x,y)=e^{-(x+y)}, \enspace x,y>0$. Find the distribution of Z defined as: $Z=\frac{X+Y}{2}...
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2answers
26 views

Given the digits of a passcode, what is the probability of getting 4 in their correct location?

Say you have a passcode $135790$ and a burglar knows exactly what digits it is made up of and that there no duplicates, but not the correct ordering. How do you calculate the probability that he gets ...
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0answers
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How to find best achievable significance level?

Given a certain $n$ number of samples, the null hypothesis states that at most 50% of those participants lean towards one answer on a survey, which is rejected only if the sample evidence strongly ...
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0answers
19 views

How D4 is calculated (Range Charts)

The formulas for the chart are found below: https://stats.stackexchange.com/questions/185373/closed-formula-for-d4-constant-calculation-moving-range-chart-constant My question is what the x1, xn ...
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1answer
33 views

Hypothesis Test when unknown mean and unknown variance

let have a random variable X which follows a normal distribution of mean $\mu$ and variance $\sigma$. we want to carry out the following hypothesis test: $H_0: \mu=\mu_0$ against $H_1: \mu\neq\mu_0$ ...
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21 views

Die roll and hypothesis testing

First of all this is not a homework or an assignment; it's an exam question that I couldn't solve. "There is a $4$-sided die that has the same probability of showing up for each face when rolled, ...
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central region of Normal distribution and Exp distribution [on hold]

This is the question in my homework i do not know how to solve it enter image description here
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1answer
26 views

The expected value of a single observation

I'm struggling to understand a line of the following proof: Claim: The sample mean $\bar{X}$ is an unbiased estimator of the population mean $\mu$ Proof The random variables each based on a single ...
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1answer
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Problem: How many 'timetables' can be created for visiting 8 different lectures without time clashes?

Apologies for the poorly worded question but it really requires more explanation than can be done with one line. I have a statistics problem, or rather, a timetabling problem that I feel can be solved ...
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Sketching residual plots

A set of data has been analysed by fitting the model $Y_i = x_i^{'}\beta + \epsilon_i$, where $\epsilon_i$ follows a normal distribution with mean 0 and variance $\sigma^2$. Sketch a residual plot for ...
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8 views

the moment generating function of the negative binomial distribution

According to the textbook I use, it states that: $X$~$Neg.bin(x;k,\theta) = {n-1 \choose k-1}\theta^k(1-\theta)^{n-k}$ Which I have no problem. The problem arises when I try to find the moment ...
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1answer
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Probability of a random subset of Z

I'm stuck in this question, could someone give me a hand? I'll post what I've done so far. Question 9: Let $A=(1,2,3,4)$ and $Z=(1,2,3,4,5,6,7,8,9,10)$, if a subset B of Z is selected by chance ...
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Consistency of maximum likelihood estimator for negative binomial distribution

Let $X = (X_1,\ldots, X_n)$, where $X_1,\ldots,X_n$ are independent and have the same distribution: $$P_{\theta}(X_i = k) = \binom{k + r - 1}{k} \cdot \theta^r \cdot (1 - \theta)^k~,$$ with $k \in \...
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1answer
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factorial moment generating function

I'm trying to get the factorial moment-generating function of a binomial random variable. I know that $F_X(t) = E[t^x] = \Sigma_xt^xp(x)$ so I get $\Sigma_xt^x{n \choose x}\theta^x(1-\theta)^1-x$ ...
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1answer
17 views

Find the density function of $(U,V)$

Let $X=(X_1,X_2)^T$ be a random vector with 2-dimensional normal distribution, $E(X_1)=E(X_2)=0 , \operatorname{Var}(X_1)=\operatorname{Var}(X_2)=1$ and $\operatorname{Cov}(X_1, X_2)= \nu$ with $|\nu| ...
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1answer
22 views

Predicting capacity problems based on historical data

Situtation: An oil-company collects waste oil from small car/truck workshops. The workshops have only limited waste oil storage varying per workshop. Oil must be collected from those workshops when ...
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2answers
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If $X_n$ and $Y_n$ are independent does $(X_n,Y_n)\overset{d}{\rightarrow}(X,Y)$?

More formally: If $X_n\overset{d}{\rightarrow}X$ and $Y_n\overset{d}{\rightarrow}Y$ and also $X_i$ and $Y_j$ are independent for all i,j; does $(X_n,Y_n)\overset{d}{\rightarrow}(X,Y)$? I am aware of ...
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0answers
10 views

False-positive in a mixture distribution

This is an example of a photoluminescence intensity signal. Suppose you have a mixture of two normal distributions for the signal data. One normal distribution is due to negative results while the ...
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14 views

To code for principal component in functional PCA [on hold]

I am trying to get the way to find eigenfunction from eigenvalues. As far as I understand from reading that I have to multiply eigenvalues by something to get principle components in functional PCA. ...
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Differential Corrections

Can you explain to me this please? How is $\chi^2$ computed with this? How is the accuracy improved by replacing $\Delta p_k$ by $\Delta p_k/2$? What is the result? $p_k$ was at the biggining, so its ...
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1answer
37 views

Finding the conditional expectation of independent exponential random variables

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Let $M = \text{min}(X,Y)$. Find (a) $E(MX|M=X)$ (b) $E(MX|M=Y)$ (c) Cov$(X,M)$ (a) I first ...
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13 views

Estimated value of sample variance for normal

Let $X_1 \ldots X_n$ be iid $N(0, \sigma^2)$ observations. I'm told that the correct answer is $E[S^2] = \sigma^2$. If I use the given value of mean in my sample variance calculation then I get the ...
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0answers
19 views

How do you create the polynomial basis function given data?

According to Wikipedia: However, i saw that basis functions applied before linear regression looks like: Is this saying that, Each observation x has D dimensions. then applying the wikipedia ...
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0answers
14 views

Using measurement scales with sets of data

When considering the four scales of measurement (nominal, ordinal, interval, and ratio), I'm wondering whether they are applied only to objects or data points, or whether they can also be applied to ...
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26 views

UMVUE of Geometric Distribution [on hold]

What is the UMVUE of $p^2$ for geometric distribution?
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Something like the Wilson confidence interval, but taking into account the “typical” success rate?

If I understand the Wilson confidence interval correctly (and I'm not confident that I do), it doesn't seem to take any data into account other than the success/failure counts of the particular thing ...
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Angle between regression line and SD line when plots flipped

The question I am working on is: The scatterplot y* vs x* has a regression line that makes an angle with SD line. Then the corresponding angle for the scatterplot of x* vs y* is: I have generated ...
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0answers
17 views

Estimating population size of endangered parrots

I am trying to find the best way to analyse the results of a study I have just coordinated. In short, I want to be able to estimate the total population size of a species of parrot based on ...