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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Predicting the number of orders from future customers

Tamara is reviewing recent orders at her deli to determine which meats she should order. She found that of 1,000 orders, 450 customers ordered turkey, 375 customers ordered ham and 250 customers ...
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I need your help for this simple statistics problem.

I need help for the following problem: In a summer reading program for youth, there is a six week period where the seven Harry Potter books are available. (1)If only three books can be read during ...
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Asymptotics of coefficients of a power series

Does anybody know what the second-order asymptotic for the coefficients of a power series means? Can you let me know where I can read more about higher order asymptotics. Thanks
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Composed distribution relation

Given independent random variables $A, B$ and $C$, for which we know $x = P(A > B)$ and $y = P(B > C)$; how, if possible, can we derive $z = P(A > C)$?
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Assuming that workers' salaries in your company are uniformly distributed between $\$35,000$ and $\$45,000$ per year

Assuming that workers' salaries in your company are uniformly distributed between $\$35,000$ and $\$45,000$ per year, calculate the average salary in your company. Please help. How to start? ...
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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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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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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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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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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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35 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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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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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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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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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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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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381 views

Variance of a MLE $\sigma^2$ estimator; how to calculate

Let $\mathbb X_1;X_2;...;X_n$ be an i.i.d. random sample from N~(0, $\sigma^{2}$). a. Find the variance of $\sigma^{2}_{MLE}$ So I found $\sigma^{2}_{MLE}$ by taking the derivative of the log of ...
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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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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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$p$-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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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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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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68 views

Variance $= 0$, show that $X=\mu$ with probability one

If the variance of $X$ is zero, show that $X=\mu$ with probability one. Using Chebychev's inequality that is, \begin{equation*} P(|X-\mu|\geq k\sigma)\leq\frac{1}{k^2}, \end{equation*} I just let ...
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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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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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Generating a given length sample and skewness whose normality is verified by one normality test but not by an other

I just want to generate 1 sample of length$=n>30$, |skewness$=S|<0.3$ and for which normality is not rejected by Shapiro wilk test of normality but rejected by Anderson darling test of ...
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What does relaxing the iid assumptions mean? Intuitive and technical perspectives.

I believe the most restrictive assumption we can place on a series of observations is that they are iid. It is possible to relax these assumptions. For example relaxing the independent distribution ...
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Independence, conditioning, and correlations

Suppose $X$ and $Y$ are independent random variables uniformly distributed on $[0,1]$. Suppose we consider a conditional distribution of $X$ and $Y$ on some event $C$. Is it possible that these ...
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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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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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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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How to calculate Fisher Information (FI) matrix for Multivariate Normal Distribution (MN)

Below is the gradient (score) of the MN log likelihood function L for n=1 observation. I originally attempted to calculate the Hessian matrix but ran into difficulty calculating 2nd order derivatives ...
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Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms…

Resistors to be used in a circuit have average resistance 200 ohms and standard deviation 10 ohms. Suppose 25 of these resistors are randomly selected to be used in a circuit. a) What is the ...
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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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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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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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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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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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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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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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How to determine the pdf for a model in phase space representation?

Consider a univariate discrete linear model : $z(k) = y(k) -(a* z(k-1) + b * z(k-2))$ where $y(k) = x(k) + \eta(k)$ $x(k) = s(k) + p*s(k-1) + q*s(k-2)$ is a Moving Average model of order 2. ...
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Mana Maximization (Hearthstone)

I recently started playing Hearthstone and a statistic / probability question came up my mind. Here's a quick breakdown: The game is a turn-based card game which involves "points" that you can used ...
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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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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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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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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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Expected value of geometric distribution

I watched $Statistics 110$ from Harvard University through YouTube. During lecture 9, I understand that the expected value of geometric distribution is $$\sum\limits_{k=0}^{\infty} ...