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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Sufficient Statistics Multi Conditional values

I am trying to find $\mathbb{E}\{X_1| X_1+X_2, X_1+X_3\}$ where all are non negative independent r.v.'s (e.g. Poisson). I am not clear about the concept of sufficient statistics, is't it enough in ...
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28 views

Method of Moments - Why is it a good estimate?

Besides the characteristic of unbiased, Method of Moments seems not as meaningful as Method of Maximum Likelihood. Why should we use Method of Moments? And, is there tools for Method of moments? ...
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23 views

Bhattacharyya Distance of Age/Gender Groups?

I'm calculating distances for groups based on Age/Gender Compositions (to rank their similarity in demographic composition.) I'm working with the following: Men 18-34, Men 35-49, Men 50-64, Men ...
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1answer
18 views

In OLS is the vector of residuals always 0? [duplicate]

I am trying to show that $$\sum_{i=1}^ne_i = 0$$ I have two hints, so to speak: $$ HX = X$$ where $H$ is the hat matrix, and that $$\sum_{i=1}^ne_i = e'1$$ My solution is as follows: $$e'1 = ...
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12 views

Statistical Dimension of a Cone

I was thinking over the following problem: Define D = {(a,b,c,d):a<=b,a<=c,b<=d,c<=d}. Let Z be a N(0,I) random vector in the 4-dimensional Euclidean Space and let Y be the projection of Z ...
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14 views

Statistics of several small datasets

I have 3 different datasets, each with 5 datapoints with a certain error. In theory, all the datapoints should have the same value, even from the different datasets (So, ideally I end up with 15 the ...
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17 views

Unbiased estimators of skewness and kurtosis

The skewness and kurtosis are defined as: $$\zeta_3 = \frac{E[(X-\mu)^3]}{E[(X-\mu)^2]^{3/2}} = \frac{\mu_3}{\sigma^3}$$ $$\zeta_4 = \frac{E[(X-\mu)^4]}{E[(X-\mu)^2]^2} = \frac{\mu_4}{\sigma^4}$$ The ...
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3answers
69 views

Intuition behind odds of winning 0 times approaching 1/e

I am learning about the number e. The wikipedia page says that in a Bernoulli experiment the odds of never winning in n trials approach 1/e as n tends to infinity. I am trying to develop an intuition ...
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1answer
32 views

Poisson Process Question relating to time between arrivals

The number of insurance claims arriving electronically in a small insurance office has been modelled as a Poisson Process with rate $5$ per hour. (I) What is the distribution of the arrival time ...
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13 views

Are variables in embedded space Statistically independent variables?

Performing Taken's phase space delay embedding on the observations $\mathbf{z}$ of a univariate random variable, with an embedding dimension $d$, we get a realization of $n$ points such as: ...
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1answer
56 views

Constructing a new Markov chain from another Markov chain

I have a very simple problem, but it seems I have difficulty to prove it rigorously. Suppose random variables $X, Y$ and $Z$ form the following Markov chain: $X\leftrightarrow Y\leftrightarrow Z$. My ...
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52 views

Poisson distribution with changing lambda

Consider two independent processes, where events occur at a constant rate (but this constant changes at a certain time). For process P1 this rate is $\lambda_1$, and for process P2 this is ...
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1answer
25 views

From pairwise P(A > B), to P(A > all distributions in set)

$\{D_0,…,D_n\}$ is a finite collection of independent (but not necessarily identically distributed) random variables. Define $f(x,y)=P(D_x≥D_y)$ and $g(x)=P(∀y:D_x≥D_y)$. Does $f$ determine $g$, and ...
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19 views

Can I adjust linear growth of a a subpopulation to a linear decay of the general population?

I need to estimate the amount of CF patients in Poland in the next four years. I have: estimations of the Polish population for the future years a CF patients' register for the last couple of years ...
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17 views

How can I use Borel-Cantelli lemmas? [duplicate]

I have this exercise , but i don't know if i can use Borel-Cantelli lemmas, or another way. " Let $X_1,X_2, ...,Y_1,Y_2,...$ independent and identically distributed random variables $\sim U(0,1)$. ...
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1answer
21 views

Expectation of cumulative distribution function of a standard normal distributed random variable

Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. The find the expectation of $\Phi(X)$. I ...
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2answers
24 views

Calculating the correlation coefficient between least square estimates

PROBLEM STATEMENT: Consider the following 2-variable linear regression where the error $e_i$ 's are independently and identically distributed with mean $0$ and variance $1$; $$y_i = α + β(x_i − \bar ...
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23 views

Expected number of rolls to roll every number [duplicate]

If I am rolling I die until I roll every number at least once, what is the expected value of times that I will need to roll the die? After a brief computer simulation, I got 15. But why is this the ...
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0answers
36 views

PDF of chi-square distribution

I'm trying to determine the PDF of Y, where $Y = \sigma \cdot \sqrt{X}$ with n degrees of freedom and $\sigma$ is a positive real #. I looked up the PDF of a chi-distribution here: ...
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14 views

Approximating energy needed to move an object with statistical information

I am stuck on following problem. It deals with energy needed to move object in lossless environment, but with lossy transmission of force that causes the movement. Imagine a vehicle moving in ...
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4answers
45 views

How to find $\sum (t-\bar{t})^2$

Given that $n=150$, $\sum t=645$ and $\sum t^2=8287.5$. How to find $\sum (t-\bar{t})^2$ where $\bar{t}$ denotes the mean of $t$.
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2answers
22 views

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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1answer
7 views

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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1answer
14 views

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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1answer
30 views

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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16 views

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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1answer
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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1answer
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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23 views

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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13 views

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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4answers
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
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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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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1answer
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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29 views

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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0answers
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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1answer
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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1answer
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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2answers
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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1answer
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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34 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, ...