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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Which statistical test measures improvement over years for different groups of people

Suppose we have a group of students taking a standardized test, as measured by a score of 1-7, and as a simplifying assumption, assume: i) Each student takes this test only once in their life ii) ...
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85 views

Method of Moments and Maximum Likelihood question

Suppose that $X_1,X_2,…,X_n$ are an i.i.d. random sample from a Rayleigh distribution with parameter $\theta > 0, f(x|\theta) = \frac{x}{\theta^2}e^{-\frac{x^2}{2\theta^2}}, x>=0$ Find the ...
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147 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
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79 views

What is the pivotal quantity

I had a question of two parts. I solved the first part but I am stuck on the second. Any hints or partial solutions would be greatly appreciated. a)$ X_1,....X_n$ are uniform iid on the interval ...
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48 views

what value for $c$ yields the estimator for $σ^2$ with the smallest mean square error among all estimators of …

If $S'^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \dfrac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S^{'2}$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the ...
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334 views

Establishing the upper and lower bounds of normal using standard deviation

I understand the concept of standard deviations and z-values, but I'm trying to figure out if standard deviations alone are good for establishing the upper and lower bounds for normal. For example, if ...
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31 views

standard deviation estimation after changing part of data to be constant

If we have a sample of known average and standard deviation, for example, 5 and 1. Suppose that we have a constant for which the CDF is 10%. The values below the constant are set to equal the ...
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31 views

some misunderstand about student distribution and R

I see on book : $t_{0.025,59}$ = 2.009 But when call qt in R. We have this result: qt(0.025,59) [1] -2.000995 I don't know why have different in here. As I think, $t_{0.025,59}$ must is ...
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23 views

Cross correlation computations

What are useful ways/formula for calculating sample cross correlations (i.e. correlation factors between individual components of two different random variables). Say I have two sample matrices, $X$ ...
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24 views

Theoretical binomial distribution fitting

I have a set of 100 randomly distributed numbers between 0 and 6(including 0 and 6) in a frequency distribution table and have worked out the mean so I can find p, however is n=6 or n=7. I'm not sure ...
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59 views

Conditional probability application problem

I was given this question in class: "A marathon is tomorrow and a watherman has predicted rain tomorrow. In recent years it has rained 101 days each year. When it actually rains the weatherman ...
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156 views

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval.

Prove that the usual (1-$\alpha$)% confidence interval for $\sigma^2$ is NOT the shortest interval. In particular, show that the minimum length interval satisfies $f_{(n+3)}(a) = f_{(n+3)}(b)$, where ...
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90 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
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205 views

show that MSE$(\hat{\theta}) = E[(\hat{\theta} − θ)^2] = V(\hat{\theta}) + (B(\hat{\theta}))^2$.

Using the identity $(\hat{\theta} − θ) = [\hat{\theta} − E(\hat{\theta})] + [E(\hat{\theta}) − θ] = [\hat{\theta} − E(\hat{\theta})] + B(\hat{\theta})$, I need to show that MSE$(\hat{\theta}) = ...
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24 views

Marginal Density Question

I am faced with the following question, which I think is quite simple, but I can't put together for some reason. Given that $f(x,y)=(6/5)(x+y^2)$ for $0<x,y<1$, ($f(x,y)=0$ everywhere else), I ...
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119 views

If $ X = \sqrt{Y_{1} Y_{2}} $, then find a multiple of $ X $ that is an unbiased estimator for $ \theta $.

Problem: Suppose that $ (Y_{1},Y_{2},Y_{3},Y_{4}) $ denotes a random sample of size $ 4 $ from a population with an exponential distribution whose probability density function $ f $ is given by ...
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15 views

Help with Poisson Random Variables.

The problem is if $\lambda=1/2$. Find $E[X]$, $E[2-X]$, $E[X^2]$ and $Var[2X]$. I know that $E[X]$ is simply $1/2$. But as for finding the other ones, I am lost. I'm assuming I'll have to create ...
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70 views

Find the distribution of $W$

Let $X \sim N(0,1)$ and $Y \sim N(0,1)$, independent. $$W = Y \;\;\text{if } Y-X >0, \;\;\;\;\;\;\;\;\; W = -Y \;\;\;\; \text{if } Y-X <0 $$ Then, find the distribution of $W$ Here is a my ...
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sum of two independent uniform random variables question

Let ܶ$T_1$ and ܶ$T_2$ be random times for a company to complete two consecutive steps in a certain process. $T_1$ and ܶ$T_2$ are measured in days and their joint probability density function is ...
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243 views

Percentage with only standard deviation and mean given. [closed]

I have some questions that I really need help with. The mean mark for an IQ test in the population is 100, with a standard deviation of 16.5. The IQ is normally distributed. Your IQ is 113. a. ...
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14 views

Did I correctly calculate the specificity and the false negative rate?

So I filled out the summary table of the data, but I'm not quite sure if I calculated the specificity and the false negative rate correctly from the table. Can someone please check that I'm doing it ...
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55 views

Take -log of a Beta distributed R.V.

X1.....Xn~Beta(a,1) Y = -log(X) Use the transformation formula to calculate the pdf of Y. What named distribution does it have? I am confused what method to use here. A beta does not converge to a ...
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65 views

Expectation formula proof [closed]

Let $X$ have a normal distribution with mean $\mu$ and variance $\sigma^2$. Prove that $E(X-\mu)^2$=$\sigma^2$
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20 views

How to decide which pdf to use when calculating expectation?

Given $Y=\ln(X)$ ~ $N(\mu,\sigma^2$), where the MLE of $\mu$ and $\sigma$ are $\mu_{hat}=1/n\sum(Y_i),\sigma_{hat}^2=1/n\sum(Y_i-\mu_{hat})^2$, respectively, or ...
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70 views

Statistics and Probability, finding unbiased estimates of mean and variance given sigma x and sigma (x^2)

The random variable $X$ is normally distributed with unknown mean $\mu$ and unknown variance $\sigma^2$. A random sample of $20$ observations on $X$ gave the following results $\sum_i X_i = 280, ...
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32 views

Conditional Variance

I'm having a bit of trouble conceptualising the step which i have highlighted.
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28 views

Hypothesis testing with P-value approach

For each trial, the individual is given a selection of 4 items to pick from labeled A, B, C, D. Each of these items is identical to exactly one of the others. (Two identical pairs, A=B and C=D, for ...
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Similarity between two XY graphs depicting the luminance of each frame of a movie, where one has noise

I've taken a sample movie A and taken a camera recording of it B. Both videos have been split into frames and the average luminance of each frame has been plotted on graphs. Original movie luminance ...
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7 views

Calculating population for significance

I am trying to rewrite the standardnormal function so I can calculate for which population n, the test is significant. z= (c1-c2)/(f1-f2)^0.5 f1+f2= (c1(1-c1))/n + (c2(1-c2))/n This is in need for ...
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10 views

control variates - estimating definite integrals

What are the good techniques to find $g(x)$ so that $f (x) - g(x)$ is minimal, in order to evaluate $\int_a^b [(f(x) - g(x)) + g(x)]dx$?
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52 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct a probabilistic model, I have tried different methods of approach, by parts, substitution, but to no avail. Any help with this would be greatly appreciated!
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37 views

Relationship between gamma and inverse gamma distributions under some algebraic operations

I have a question about the relationship between gamma and inverse gamma distributions. I have an equation that takes $L/(c*X)$ Where $X$ is a Gamma distribution and $c$ and $L$ are constants I'd ...
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61 views

Inverting probability generating function via mellin transform substitution.

The pgf is defined as: $E(z^k)= \sum_{k=0}^{\infty} p(k)z^k$ which is a discretised version of the transform: $\widetilde{p}(z) =\int_{-\infty}^{\infty} z^k p(k) \, \mathrm{d}k$ The Mellin ...
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37 views

Proof of statistics equality [closed]

Equality is: $$ \left(\sum_{i=1}^{n} (Y_{i} - \overline{Y})\right)^{2} = \sum_{i=1}^{n} (Y_{i}^{2} - n\overline{Y})$$ Please explain why equality holds.
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Probability of being selected in a raffle

There's a raffle with 1,000 names in a bucket. 600 of those names are in there once, and 200 are in there twice. So, just to reiterate, there are 800 unique names in the raffle, and 1000 names total. ...
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17 views

Hypothesis testing question trying to find the p value

for each of the data sets find the p value This is what I have so far ts = (735-854)-0/38 = -3.13 |ts| = 3.13 I am having a hard time getting the P-value. Please help.
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121 views

How do I measure the goodness of cosine similarity scores across different vector spaces?

I am a computer scientist working on a problem that requires some statistical measures, though (not being very well versed in statistics) I am not quite sure what statistics to use. Overview: I ...
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24 views

Finding p-value when df and test stat is given. PLEASE HELP.

For a chi-square goodness of fit test with 10 degrees of freedom, the test stat is 20.000, then how to find the p-value? In other words, what is ?: X^2?,10 = 20.000? The X^2 table does not give the ...
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55 views

How do I find $\theta$ with bootstrap?

I have two vectors of known values $x$ and $y$. And the relationship between them is $y=\sin(\theta \cdot x)+\epsilon$, $\epsilon \sim N(0,1) $ . The question is how do I estimate $\theta$ with ...
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55 views

Probablility of a Dice Game

Player A rolls $m$ dice, while Player B rolls $m + 1$ dice. If Player A rolls $a$ $n$'s and Player B rolls $b$ $n$'s, then Player A wins if $a > b$ . Otherwise, Player A rolls up to $k$ of the $m$ ...
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115 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
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How to find MLE function?

I have two vectors of known values x and y. And the relationship between them is y=sin($\theta$$\cdot$x)+$\epsilon$, $\epsilon$~N(0,1) . The question is how i find the MLE function for $\theta$?
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Generate correlated random numbers precisely

Let's assume I want to generate k samples of n random numbers, that are correlated according to a given correlation matrix C (e.g. $n = 3$): ...
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21 views

proving a complete sufficent statistics

Suppose that $ X_1,\ldots,X_n$ are iid poisson($b$); $c = b^2$ and $S_n = \sum X_i$ To Show that $S_n$ is a complete sufficient statistic for $c$. I can prove using exponential family that $S_n$ is ...
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36 views

MLE of discrete uniform distribution

Assume that $X$ is a discrete random variable with uniform distribution on the set $\{1,2,3,\ldots, N\}$, where $N$ is an unknown positive integer. Find the MLE $\hat{N}_k$ of $N$, assuming that ...
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52 views

Uniform Distribution and Distribution function technique

Let $X_1$ and $X_2$ be independent random variables having the uniform density with $\alpha = 0$ and $\beta = 1$. Find expressions for the function $Y =X_1 + X_2$. (a)$y \le 0$ (b)$0<y<1$ ...
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Statistics Primer for the Unwary Mathematician

I have a new position in a biology department (after being housed in a maths department) working on cognitive and population modeling. People in my lab are asking for help with applying statistical ...
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99 views

Distribution function technique and exponential density

I'm having quite a bit of difficulty with the distribution function technique. If $X_1$ and $X_2$ are independent random variables having exponential densities with parameter $\theta_1$ and ...
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48 views

a question which is somhow related to law of large number

suppose that $\mathbf p = [p_1, p_2, ..., p_n]'$ is a random vector. (' == transpose) and each element of $\mathbf p$ like $p_i$ is a Gaussian random variable with zero mean ($\mathbb E(p_i)=0$) and ...
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Multiple measurements per person per treatment

Suppose I wish to assess reaction time of individuals before and after treatment. Now to analyse the results I could use a paired t-test or if I had additional treatments, I could use a repeated ...