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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confidence and estimating

You wish to estimate,with 99% confidence, the proportion of Canadian drivers who want the speed limit raised to 130 kph. Your estimate must be accurate to within 5%. How many drivers must you ...
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31 views

Is this an improper method of averaging grades? If so, what is a simple mathematical way of explaining it?

I have a professor who employs a unique method of averaging grades. On each assessment, the professor assigns a raw numerical score to each student based on performance. He then converts particular ...
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31 views

Confidence and proportion

You wish to estimate,with $99\%$ confidence, the proportion of Canadian drivers who want the speed limit raised to $130$ kph. Your estimate must be accurate to within $5\%$. How many drivers must you ...
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22 views

Comparing Expected vs Observed with almost no information

I have a game with somewhat intrincate rules regarding its prizes (it's vide-bingo). Thankfully, we managed to find out the expected mean $\mu_0$, for the liniarity of the Expected Value. Even though, ...
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seperating a 100m in a 100 pieces 1 meter at a time with $X_1,X_2,..,X_{100}$ the errors of each measurment

The problem is the folowing: we want to seperate 100 meters in a 100 pieces we do this by measuring 1 meter at a time. Let the errors made in each measurment be $X_1,X_2,...X_{100}$ and i.i.d with ...
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38 views

From sample mean and variance of $X$ to $\sqrt{X}$

I have samples $x_i$ of lets say a random variable $X$ (euclidean distances, $X=\sqrt{Y}$, where $Y$ is the squared distance) which I computed from squared distances samples $y_i$. I can now calculate ...
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26 views

Standard Error of the Mean

I have a basic question. Calculate a $95\%$ confidence interval for the mean where: $S= 1.25$ $\overline{x} = 1.14$ $z = 1.96 $ $n = 250$. My understanding is that you use the following ...
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85 views

How to prove it is a strictly stationary process?

$ξ(t) = z*sin(ωt + θ)$ where $z$ is a random variable and its distribution is unknown and $θ$ is another random variable that is independent of $z$ and $θ$ is uniformly distributed on $(0, 2\pi)$. ...
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25 views

Looking for a “Black-Scholes-esque” expression for $E[\max(V-K,Y)]$

In Hull (2008, p. 307), the following equation is found (Eq. 13A.2): $$E[\max(V-K,0)]=\int_{K}^{\infty} (V-K)g(V)\:dV$$ Where $g(V)$ is the PDF of $V$, $K$ is a constant, and both $V,K>0$. He ...
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5answers
34 views

Evaluating $E[\max(X,Y)]$

Let X and Y be positive independent random variables, and $$W=\max(X,Y)$$ Define the CDFs of X and Y as $F(x)$ and $G(y)$, respectively. $$\Pr(W\le w)=\Pr(X\le w)\Pr(Y\le w)=F(w)G(w)$$ ...
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36 views

Probability of multiple dice rolls with decreasing amounts of dice

Calculating probabilities over multiple dice rolls is easy, but what do you do if the amount of dice decreases (dependently) from roll to roll? This is a common feature of many games, including Risk, ...
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22 views

Non-Whole Median Numbers in Real Data [duplicate]

According to this CDC report, the median number of reported sexual partners for females aged 15-44 is 3.2, and for males 5.1. Tables on pages 19 and 20 report these statistics for a variety of ...
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18 views

Finding Bayes risk

I have that $f(x;\theta)=\frac{e^{-x}x^\theta}{\theta!}, x>0$ and $\pi(\theta)=(1-\alpha)\alpha^\theta, \theta=0,1,2,...$ with $0<\alpha<1$ where $\alpha$ is a known hyperparameter. I've ...
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39 views

Calculating MSE of the estimate $T=\max\{X_1,X_2,\ldots,X_n\}$ of $\theta$.

The variables $X_1,X_2,\ldots X_n$ are i.i.d uniform distributed on $[0,\theta]$. $$T=\max\{X_1,\ldots,X_n\}$$ is the estimate of $\theta$. I need to calculate MSE. I know that ...
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3answers
75 views

Which law in probability theory states the following?

Which law in probability theory states the following? If we have a large enough number of samples, their histogram function converges their true probability density function. (for a continuous ...
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11 views

Normalizing Data for thickness

Math is not my strong point and I am struggling with trying to figure out how to solve the following problem...any help you can offer will be greatly appreciated! I'm looking to normalize this data ...
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How does REML estimation works?

I am trying to understand REML estimation for variance. So far I have been able to understand the obvious advantage of using it instead of maximum likelihood estimation(MLE). But I wanted to ...
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33 views

how to solve the example on reject or accept the claim

A lady stenographer claims that she can take dictation at the rate of 118 words per minute can we reject her claim on the basis of 100 trials in which she demonstrates a mean of 116 words and a S.D. ...
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How does a median have a value that is a decimal which isn't exactly half of an integer if the data should consist of only integer values?

I real an article which said the average man accumulated 6.1 sexual partners while the average woman accumulates 3.6. If the statistic talked about the average, surely the numbers would be equal-so it ...
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38 views

If a 17%-efficient system becomes “10 times more efficient”, what is the absolute efficiency? Or is this not possible?

Sometimes in reading around the net, I see things like "This car could be ten times more efficient if the drivetrain and engine were replaced by batteries and electric-motor wheels." If I'm not ...
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9 views

Prediction intervals with OLS and indicator variables

Suppose I have a model like so, call it the first model: $$E[y] = \beta_0+\beta_1x+\beta_2x_m+\beta_3(x\cdot x_m) $$ where $x_m$ is an indicator variable. I fit it using ordinary least squares. ...
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2answers
30 views

Teasing apart an explanation of the Central Limit Theorem

I'm looking at the central limit theorem, and cannot see in the explanation given to me how the average of identical distributions results in the normal distribution. I am told to consider a sequence ...
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55 views

Outlier detection with robust multiple regression model

I have a set of features (eg, location, income, budget, education) that I use to predict a continuous variable (say, amount spent per day on the internet). I am interested in detecting outliers. I ...
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42 views

Comparison between maximum likelihood and least square methods.

I understand the maximum likelihood and least square methods individually for parameter estimation. It appears maximum likelihood is very general and least square solution is applicable for a class of ...
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39 views

Distribution proportional to size

This is almost too basic: Let $X$ be a discrete random variable with support $\{1,2,...,n-1\}$ such that $$P(X=k)=k/N,$$ where $N:=\binom{n}{2}$. Does this distribution have a common name? It's also ...
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30 views

Mean estimate and Least square estimate.

This question is refers to the parameter estimate by average value given by the link: https://en.wikipedia.org/wiki/Mean and least square estimates by the link: ...
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38 views

What is the best way to interpolate over the 25th and 75th percentile of SAT scores?

The problem: I know the 25th and 75th percentiles of SAT scores for students admitted to a given university, and I want to interpolate over those two points in order to estimate all the percentiles ...
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2answers
31 views

Sampling from the von-Mises Fisher distribution?

This topic has already been tackled on this website (here). But, unfortunately, no clear cut answers were given. In (Wood,1994), there is apparently a rejection algorithm for sampling from this ...
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20 views

Likelihood of two biased coins having same heads probabilty

Say I have two biased coins of which I don't know their respective heads probability - let's call those $p_1$ and $p_2$. Say I launch them both (independently) $n$ times, obtaining $s_1$ and $s_2$ ...
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13 views

Re-writing exponent of Multivariate Gaussian

In Bishop's Pattern Recognition and Machine Learning (ISBN-13: 978-0387-31073-2), Bishop writes on page 86: This is an example of a rather common operation associated with Gaussian ...
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Determining density function of continuous random variables

Let X be a continuous random variable with the following density function $$ \ f(x) =\left\{ \begin{array}{ll} 2x^{-2}\:for\:x \geq 2 \\ 0 \: otherwise \end{array} ...
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Do we only use T distribution for confidence interval of Beta for linear regression or we can also normal distribution?

Do we only use T distribution for confidence interval of Beta for linear regression or we can also normal distribution? Is it that when sample size is less than 30 then we use T distribution else ...
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Sum of waves with random phase and amplitudes as random sum of cosines

I need to derive the average and variance of the amplitude of a sum of waves with the form: $$ \sum_{k=1}^N e^{j\delta_k} A_k $$ where $$A_k = \sum_{i=1}^N \cos(\phi_k - \phi_i)$$ The random ...
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Variance of the MVUE for a geometric parameter

I am trying to find the variance of the MVUE (built with Rao-Blackwellisation) of the estimator of the parameter $p$ of success in a Geometric distribution counting the number of failure. It seams to ...
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prove : if E(X) doesn't exist $E(x^2)$ too doesn't exist.

$E(X^2) $ exists implies $\int x^2 f_X(x) \ dx < \infty$ now from the property of Riemann Integral $\int|x| f_X(x) \ dx \le \int x^2 f_X(x) \ dx $ . hence, existence of $E(X^2)$ implies ...
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Why a normal distribution is sufficient?

Consider a random sample from a normal distribution that has an unknown mean and variance. We have $$f(\mathbf{x}\mid\mu,\sigma)=\prod^n_{i=1}\frac{1}{\sigma\sqrt{2\pi}} ...
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How will law of large number changes if we have Indepedent but not identically distributed?

How will law of large number changes if we have Indepedent but not identically distributed ?
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How many elements use for device to work with certain probability.

Hello I have exercise like this: Some device is build from $n$ elements. Device works if at least 97% of elements is working. Probability of each element breaking is $0.02$ . How many elements must ...
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70 views

Find the distribution of linear combination of independent random variables

Given independent and identically distributed random variables $X_1, X_2, \dots, X_n$, each of them has the same p.d.f $f(x) = Pr(X = x)$ on support $(a, b)$. How do I find the pdf or cdf of $Y = ...
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39 views

Sampling with no duplicates

I am sampling a population of unknown size and unknown distribution. The sample will be taken over distinct time intervals, but I have to reject any duplicates in the given time interval. The sample ...
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16 views

Method of coding to find the mean and standard deviation

The mass of each of 300 packets of flour produced by a factory in labelled 1kg. When the mass of each packet is measured to the nearest $0.001$ kg, the following frequency distribution is obtained. ...
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40 views

Realisations of i.i.d. random variables with a density are linearly independent over $\mathbb{Z}$

I need to use the fact that for i.i.d. random variables with a density $X_1, X_2, ... , X_n$ the set of outcomes $\{x_1,x_2,...,x_n\}$ are linearly independent over $\mathbb{Z}$ with probability 1, ...
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24 views

Statistical/Combinatorial: How to analyze?

I'm currently preparing for my exam and in the process trying to solve some statistical problems. The question goes as follows: Q1: A book consisting of 269 pages contains 40 missprints. Only, you ...
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Momenent generating function of a transformation.

Let $X \sim \mathrm{Poisson}(\lambda)$ so the probability function is, (for some $\lambda > 0$) $f_X(x)=P(X =x)= \dfrac{\lambda^{x}e^{-x}}{x!}$, $x=0,1,2,\ldots$ the moment generating function of ...
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why does the squared norm projection of a standard normal follow a chi squared distribution

Let $Z \sim \mathcal{N} ({\bf 0}, I_d)$ be a $d$-dimensional multivariate normal RV, $n$ be a unit vector where all components are non-zero, then $Z$ projected on to the plane normal to $n$ is $Z - ...
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Density function of a transformation

A random variable $X$ has density $f_{X}(x)=2x$ ($0 < x <1$). Find $f_{Y}(y)$ for $Y = X^{3}$. How is the density function calculated for $y$ in this case? Are we simply cubing ...
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2answers
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$E(X+Y)$ of independent bivariate distribution.

Let $X$ and $Y$ be random variables with joint density function $f_{X,Y}(x,y) = \frac{1}{100}e^{\frac{-(x+y)}{10}}$, $x>0$, $y>0$. Calculate the expected value $E(X+Y)$ Given I know the ...
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26 views

Proof arithmetic mean is bounded

Suppose $A$ is the arithmetic mean of the set of real numbers $S$. How would I go about proving that $$\min<A<\max$$ Where $\min$ and $\max$ are the minimal and maximal element of $S$, ...
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40 views

Statement concerning Poisson random variables that appeared in a proof I am studying

So I am studying a proof and the following line appeared. Assume $X_1,...,X_n,X_1'...X_n'$ i.i.d. Poisson random variables of mean $\frac{\mu}{n}$ each, and $f$ a real valued function such that $|f| ...
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Rao-Blackwell Theorem corollary

I have as corollary to the Rao-Blackwell theorem: If a minimum variance unbiased estimator $\hat \theta$ for $\theta$ exists, there is a function $\hat \theta_T$ of the minimal sufficient statistic ...