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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Elliptically symmetric random variable

Let $X$ is elliptically symmetric distributed random vector. Then $X$ can be expressed in the form $$X =^d \mu + R A U$$ where $R$ is a nonnegative random variabel and $U$ is uniformly distributed ...
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21 views

Autocovariance Function

I need some help please Let $Y_t$ be stationary zero-mean process. Consider the model $X_t=(1-0.4B)Y_t$ How I find the autocovariance generating function of $X_t$? I multiply both sides by ...
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10 views

Dynamic weighed probability

If I have a list of objects, and want [programmatically] to randomly pull one out, I can simply choose a random number representing a valid index within the range of elements of the list: ...
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what do z, p, phi mean in statistics

What do these mean? I have found them in a paper but I am not exactly sure how to understand them. $z=2.46$, $p < .02$, $p_{rep} > 0.92$, $\phi = -.14 $
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Nonparametric changepoint detection for point process

This is a replication of a question I've recently asked on Cross Validated. It hasn't received an answer or much attention, so I've posted it here. I have a family of point processes representing ...
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62 views

I don't know what is T in this exercise

Let $X_1, X_2, \dots$ independent and identically distributed random variables $\sim Bernoulli(p)$. $\ T = \inf (n : X_{n-1}+X_{n}=1)$, calculate 1) P(T=n) ; 2) E(T). But I don't know how to resolve ...
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41 views

Line Of Regression given x? [closed]

You have found the regression line for a set of data points to be: ŷ = 30.23x + 173.52. Use the line to predict the value of y when x = 48.
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1answer
36 views

Maximum variance of a discrete probability distribution over the non-negative integers.

Suppose $P_{n}$ is a probability distribution over the non-negative integers (i.e. $n=0,1,2,...$). Also, assume that the average \begin{equation} \bar{n} :=\langle n\rangle= \sum_{n=0}^{\infty} n \, ...
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13 views

Estimate Sample [duplicate]

You wish to estimate, with 99% confidence, the proportion of drivers who want the speed limit raised to 130 kph. Your estimate must be accurate to within 5%. How many drivers must you survey, if your ...
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1answer
14 views

Point Estimates Using C.I.

0.680 < p < 0.800 What is the point estimate for p, and the margin of error from which the C.I. was formed? I am confused as to what "p̂" and "E" are equal to. Normally, I would use the ...
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23 views

Measuring incoming communication in a Markov Model

Given a standard Markov Chain on discrete time and finite statespace, represented by a matrix $M$, with $\sum_{j=1}^d m_{ij}=1$. I have a certain absorbing state k, where the incoming communication ...
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Hypothesis Claim

A company claims over 60% of dvds stop working within 2 years, and you must test this at the $0.05$ level of significance. State the claim and counterclaim mathematically, and Label which is $H_o$ and ...
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$p$-value in hypothesis testing

Find $p$-value, make appropriate conclusion about $H_0$. Left tailed test ($H_a$ is $<$), $z= -1.28$, $\alpha= 0.05$ Two-tailed test ($H_a$ is $\neq$), $z= 1.28$, $\alpha=.01$ ...
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1answer
31 views

How to calculate expected value of normal distribution with the condition that value is higher than x

I have following problem. Let assume that lifespan in the population has normal distribution with certain mean, variance and skewness. When the baby is born, its average lifespan will be equal to ...
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1answer
24 views

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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1answer
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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24 views

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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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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1answer
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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
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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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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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1answer
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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38 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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1answer
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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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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1answer
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
29 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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41 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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1answer
29 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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1answer
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
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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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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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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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2answers
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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}} ...