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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Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
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Prove that a symmetric distribution has zero skewness

Prove that a symmetric distribution has zero skewness. Okay so the question states : First prove that a distribution symmetric about a point a, has mean a. I found an answer on how to prove this ...
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Is the Monte Hall Problem a Ludic Fallacy? [on hold]

---- EDIT ---- My Questions: 1) Does cheating mathematically reduce your odds? 2) Regarding the Monte Hall Problem, how is it any different than 'cheating' when a host helps you? (Original very LONG ...
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56 views

Expected value of the larger of two claims

Claim amounts for wind damage to insured homes are independent random variables with common density function $f(x) = \frac{3}{x^4}$ for $x > 1$; and $f(x)=0$ otherwise. Suppose two claims are ...
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Find lower bound of probability value using Chebyshev's inequality

Given density function of random variabel $X$ is $f(x) = 3x^2$, for $0 \lt x \lt 1$. Use Chebyshev's inequality to find lower bound of probability value : $P(5/8 \lt x \lt 7/8)$ $P(1/2 \lt x \lt 1)$ ...
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27 views

What statistical equation am I looking for?

Suppose you have two data sets: Dataset $A$: $4$, $2$, $6$, $5$, $3$, $3$, $5$, $4$, $4$ Dataset $B$: $8$, $0$, $0$, $8$, $9$, $1$, $1$, $9$, $0$ Both data sets have a mean of $4$, but dataset ...
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Mean & SD of Sampling Distribution

A population consists of $4$ numbers $\{0, 2, 4, 6\}$. Consider drawing a random sample of size $n = 2$ with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal ...
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112 views

What to do with the boundary values of a bin in a histogram?

Suppose I want to make a simple frequency histogram of the following data: $$\{3, 3, 4, 5, 5, 6, 7, 7, 8, 10, 11\}$$ I'm supposed to use bins of size $5$, starting with zero. Here's my question: Is ...
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137 views

Calculating the variance, mean, and autocorrelation of a time series.

How can I calculate the mean, variance, and autocorrelation function: $$Y_t=5+Z_t+ 0.6Z_t-1$$
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25 views

What does “central value” mean?

How to calculate central value of the following sets: I'm thinking is the same as the median - is it? $\{-2, -1, 3, 5, 7 , 1, 3 , 6, 2 , -1, -5\}$ and $\{-2, -1, 3, 5, 7 , 1, 3 , 6, 2 , -1 \}$
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Expectation of a discrete variable [on hold]

Suppose $P(X=n)=kq^{n-1}$ where k is a constant. Then how do I find a value for the expectation of X?
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Random Variables and Statistic

I'm studying Statistical Inference by Casella and I'm confused with the definitions of random variable & statistic. So let we have the probability space $(\Omega, F, P)$ where $\Omega$ is the ...
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3answers
18 views

Applying the basic formula for binomial distribution

I'm pretty confused on how this works. In my class my teacher states that: Let $X$ be a random variable with $S_X = \{0,1\}$. $X$ follows a Bernoulli distribution if $P(X = x) = p^x(1-p)^{1-x}$ for ...
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Proving an inequality involving uniformly distributed random variables

$X_1, \ldots, X_k$ are uniformly distributed random variables on the interval $[0,1]$. With $Y_{k-1,i}$ we denote the $i$-th smallest nuber in $\{X_1, \ldots, X_{k-1}\}$. How can I prove the ...
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1answer
11 views

Find the probability P(x is even) of given cumulative distributive function [on hold]

Given cumulative distributive function (CDF) $F(x) = 1 - (1/2)^{(x+1)}$ for $x = 0, 1, 2, ...$ Find the probability value P(x is even).
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Find upper bound of probability value using Chebyshev's inequality [on hold]

Given density function of random variabel X is f(x) = 1/(2√x), for -√3 < x < √3. Use Chebyshev's inequality to find upper bound of probability value P(IxI≥3/2).
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68 views

Profile likelihood: Box-Cox transformation

I'm trying to prove a result that shows that the maximum likelihood estimator reduces the number of parameters in a Box-Cox model. In essence, we're trying to prove that $\bar{z}$ is the nuisance ...
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16 views

expectation calculation problem small problem

a Continuous, positive random variable X, whose PDF is proportional to $(1+x)^{-4}$, where $0<x<\infty$, determine $E(X)$ i tried to solve it directly by integrating from 0 to infinity to get ...
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Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
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3answers
420 views

integrating using student t distribution

Evaluate the integral $\int_0^\infty\frac{1}{1+x^2}dx$ using the Student t distribution. I don't know where to start. I am assuming that I can't just do regular integration. I don't know how I am ...
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2answers
23 views

expectation calculation problem

I got the answers for this and i know its 1.05 but the way it explains is very difficult to understand so im seeking for some help here. A system made up of 7 components with independent, identically ...
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1answer
42 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...
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1answer
54 views

incorrect rejection of a true null hypothesis?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
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Distribution of random variables (normal and standard normal)

Suppose that $X_i \sim N(\mu, \sigma^2)$ for $i = 1, \ldots, n$ and that $Z_i \sim N(0,1)$ where all of the random variables are independent. Denote $s^2_Z$ as the sample variance of $Z_1 , \ldots, ...
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4 cards are shuffled and placed face down. Hidden faces display 4 elements: earth, wind, fire, water. You turn over cards until win or lose.

Question: 4 cards are shuffled and placed face down in front of you. Their hidden faces display 4 elements: water, earth, wind, fire. You turn over cards until win or lose. You win if you turn over ...
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33 views

How to find $E[Y|X=1]$?

A fair die is repeatedly rolled. Let $X$ and $Y$ denote, respectively, the number of rolls required to obtain a $1$ and a $2$. How do I find $E[Y|X=1]$? edit: for using this I got 1*6 = 6 total ...
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1answer
31 views

formal proof that p-values are uniformly distributed

I'm trying to prove that $p$-values under the null hypothesis are uniformly distributed in $[0, 1]$ for an absolutely continuous test statistic $X$. Proof: By continuity of $F_X$, it is sufficient to ...
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1answer
307 views

Confidence interval of quotient of two random variables

I have random variables $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$, with $n$ a large integer. All pairs $(X_i, Y_i)$ are independent and identically distributed, but every $X_i$ and $Y_i$ ...
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Prove by induction: $E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$ Please just check what I've done

Prove by induction: $$E[\sum_{i=1}^nc_iU_i(X)]=\sum_{i=1}^nc_iE[U_i(X)]$$ Let me show you what I've done. I think I'm right: $$n=1,$$ $$E[c_1U_1(X)] = c_1E[U_1(X)]$$ Okay so maybe this one looks ...
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Standard deviation and mean question [on hold]

The pulse rates of healthy adults are normally distributed with mean 72 and standard deviation of 5. i) Approximately ninety-five percent of the adult population can be said to have their ...
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51 views

Two independent Poisson processes.

I am trying to prove the result that exactly $k$ occurrences of a Poisson process before the first occurrence of another independent Poisson process is a geometric random variable. \begin{align} ...
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A mix between the Horvitz-Thompson and ordinary estimator

I have asked this question on mathoverflow, but got no answer. Here I have corrected some mistakes and wish to hear any ideas that may bring at least numerical result: The data I have two samples: ...
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1answer
16 views

Selection of Distribution model

An expressed parcel delivery company offers a First Class service for which it is promised that 80% of all parcels are delivered within 24 hours of dispatch. It is suspected that the true successful ...
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22 views

Deriving a formula for a confidence interval

Derive a formula for a $(1-\alpha)100\%$ C.I. for $\mu_x -\mu_y $ for data that has the following properties: A random sample $X_1,X_2...X_n \ are \ i.i.d ~N(\mu_x, \sigma^2 ) $ Another random ] ...
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What is the distribution of the euclidean distance between two normally distributed random variables with non-zero means?

Assuming two uncorrelated random variable (RVs) with Gaussian distributions $x\sim N(m_1,s)$ and $y\sim N(m_2,s)$, so with non-zero mean and same variance, what is the distribution of $z=\sqrt{(x^2 + ...
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How to get more profit in stochastic process?

Suppose there is a system, for each step, I cost something but I didn't know how much I cost, and the system return to me something, which follow Guassian distribution and the expectation is what I ...
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How to prove expected value of uniform random variable?

I tried this: $$\int_a^b t~dt = \frac{t^2}{2}\Big]_a^b = \frac{b^2-a^2}{2} = \frac{(b+a)(b-a)}{2}$$ Isn't it supposed to be $\frac{b+a}{2}$ or something like that? Obviously if I multiply the ...
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computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
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22 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
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1answer
25 views

Method of moments estimation for $\theta$

I read one example in my notes, but I couldn't find out how the answer in my notes is derived. If $x_1,...,x_n$ are realizations of a random variable distributed with the following PDF: $f(z; ...
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1answer
28 views

Comparison of parameter: two different populations

I was wondering what the best way is to check for the equality of two parameters for a regression with no constant including possibly a confidence interval and p-value. $$H_0:\beta_1=\beta_2\ \vert\ ...
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27 views

How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
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F-test and T-test produce the same results

I am modelling a stochastic process by two different methodologies and I expect the results of each to be normally distributed with identical means and stdevs. To test that the distributions after ...
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Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
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Argmax distribution of Brwonian motion plus linear drift

I want to know the the density function or the tail of the density funcion of the following random variables: $$\underset{{t\in [0,+\infty]}}{\arg \max} \quad {W_t-t}.$$ Thank you very much
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1answer
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Uniformly distributed independent random Variables [on hold]

Let X and Y be independent random variables each uniformly distributed on (0,1). Find $P(Y\geq X | Y\geq \frac{1}{2})$. The answer is $\frac{3}{4}$ But I don't know how they got it :( Please help as I ...
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1answer
287 views

How do I compute “AUC” Area under the curve number, if all I have are my TPR and FPR values?

I am trying to rank my neural network, which is trained for binary classification. That is, given a set of input signals, it outputs either a 1 or a 0. I have a training set, where I have the actual ...
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20 views

Calculate P-Value

In a certain area, regulations require that the chlorine level in wastewater discharges be less than 100 $\mu$/L. In a sample of 85 wastewater specimens, the mean chlorine concentration was 98 ...
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You are making cookies and add N chips to dough randomly, and split it into 100 equal cookies, again at random. How many chips should go into dough?

Question: You are making chocolate chip cookies. You add N chips randomly to the dough and you randomly split the dough into 100 equal cookies. How many chips should go into the dough to give a ...
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31 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...