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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maximum likelihood estimator for theta

I was wondering if someone could please just get me started on this question i'm just a bit stuck: $$ f(y_1,y_2,\ldots,y_n\mid \theta)\propto \exp\left[\frac{−1}8 \sum_i (y_i−\theta)^2\right] $$ Any ...
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Possible to eliminate mutual information between random variables by reducing the number of them?

Say you have a set of random variables that have some mutual information structure. Could be that they all have nonzero MI between them. Or perhaps there are some clusters of variables with ...
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8 views

Statistical Dependency Transitivity

I came across this question here on Stack Exchange, and it didn't address something that I then became curious about. If $X_1, X_2$ are dependent and $X_2, X_3$ are dependent, then are $X_1, X_3$ ...
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7 views

Is ergodic in the mean process the same as wide sense stationary?

I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows: General Moving Average random process given as $X[n]=(U[n]+U[n-1])/2$ where $E[U[n]]=\mu$ and ...
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1answer
29 views

Moment generating functions…which distributions to use?

Q: You hired a terrible programmer and the moment generating function for the distribution of software bugs is M(t) = (1 - $\theta$t)$^{-\alpha}$. Groups of bugs can be detected within $\mu$ = 47 ...
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I have some questions related to Fisher's book 1925.

I was studying Fisher 1925 and while reading i had some trouble with this part. Fitting the Normal Distribution From a sample of $n$ individuals of a normal population the mean and the standard ...
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1answer
27 views

Is tossing a die in 10 consequent days an ergodic process?

IT maybe an elementary question but I'm totally new to the concept. In Wikipedia, ergodicity is defined as follows: In statistics, the term describes a random process for which the time ...
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1answer
13 views

what's the difference between variable and process from a statistical point of view?

I'm reading a tutorial stochastic process: ergodicity and temporal averages and I'm totally confused. It is said that: Suppose an IID random process whose marginal PDF is Gaussian with mean ...
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16 views

Ensemble average of square of fluctuations proof

The ensemble average of a random variable $x$ is denoted as $X$ or $\left \langle x \right \rangle$, and is defined as: $$ X = \left \langle x \right \rangle = \lim_{N \to \infty} \frac{1}{N} ...
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23 views

Regression in Bivariate Normal

Suppose $(X_i \hspace{4pt} Y_i)'$ are $i.i.d$ $N_2 (\bf{\mu,\Sigma})$, $i=1(1)n$ where $E(X)=\mu_x$, $E(Y)=\mu_y$ and $\Sigma$ is given by \begin{bmatrix} \sigma^2_x & \rho\sigma_x\sigma_y\\ ...
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Comparison of Cramer Rao bound - deduction and conceptual question

The CRB gives the variance of the estimation error of the estimates and a lower value is preferred. I have computed the cramer rao bound (CRB) of the estimates of the coefficients $\mathbf{h^T}$ for ...
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35 views

what is the difference between statiscal averagre and average?

I'm reading a book on synthetic aperture radar and it is said that: The term $\sigma^{\circ}$ is the averaged radar cross section per unit area, also called the scattering coefficient or ...
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1answer
15 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
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1answer
13 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
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42 views

A good, self-study statistical computing book

I'm looking for a book an introductory statistical computing that has proofs for the methods as well as examples. I'd like proofs that are about the same level as (or lower than) proofs in Statistical ...
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1answer
29 views

Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
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24 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
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1answer
29 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
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2answers
34 views

What is the variance of the volumes of particles?

According to Zimmels (1983), the sizes of particles used in sedimentation experiments often have a uniform distribution. In sedimentation involving mixtures of particles of various sizes, the larger ...
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31 views

Class Coin Toss Experiment

My classmates and I are doing a coin toss experiment (i.e. toss coin 100 times). I have already determined that I have a fair coin, since I tossed $43$ heads, and this falls into a $95$% confidence ...
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What is the proof behind the mean confidence interval for a Binomial Distribution?

How do we obtain the range to be as [$\mu-$$zσ$, $\mu+$$zσ$]? Is it when $n$ is sufficiently big?
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1answer
29 views

Pairing birthdays [on hold]

How large a group of people would you need to provide a better than 50-50 chance that everyone will have at least one birthday (just month/day) partner?
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23 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
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1answer
37 views

Cramer-Rao lower bound for normal($\theta, 4\theta^2$)

I am trying to find the Cramer-Rao lower bound for unbiased estimators of $\theta$, given a sample $X_1,\ldots, X_n \sim \textrm{normal}(\theta,4\theta^2)$. I am calculating the CRLB as $$ ...
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1answer
59 views

How to make statistical sense of this experiment:

I have conducted an experiment but I am now unsure of how to say, from a statistics point of view, that the data supports or not that a certain phenomenon has occurred, meaning it could be mere ...
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1answer
18 views

sampling distributions and test of hypothesis

A manufacturer of a certain type of breakfast cereal claims to produce packets which contain on average 500 grams of cereals. Ten packets were selected at random and the cereals content of each ...
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25 views

Distribution of the test statistic?

Let $\mathbf{x}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma)$. I am trying to find a distribution of the following test statistic $ T(\mathbf{x}) = \frac{\bar{\mathbf{x}}^H ...
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Meaningful Extreme value distribution

Extreme value theory (EVT) dictates that the limit distribution of the minimum of the set of i.i.d. Chi-square random varibales $\{C_1,C_2,\cdots,C_n\}$ is Weibull. The Weibull distribution has ...
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1answer
18 views

Which is the probability?

On the Swedish SAT test, you have 5 options for every question where precisely one option is correct. If you answer randomly, what is the probability that your score will be 0.9 if the maximum score ...
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1answer
31 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
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0answers
14 views

Help in calculating the Hessian Matrix from the log-likelihood

I am trying to find the Fisher Information Matrix for a univariate linear linear Moving Average model: \begin{align} z(n) &= h_1 u(n-1) + h_2 u(n-2) + u(n) \tag{1} \\ y(n) &= \mathbf{h^Tz(n)} ...
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1answer
40 views

How to summarize a big table of results? Average or Geometric mean?

I am writing a paper for a Computer Science conference and I have a big (way too big) table of results (times and some other measures) for different versions of an algorithm. I would like to summarize ...
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1answer
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In statistics using a regression analysis in SPSS - - variables are hunger and amount of dancing

In statistics using a regression analysis in SPSS - - variables are hunger and amount of dancing. Which would be the dependent and independent variables? Thanks!
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2answers
28 views

Distribution of a product of Multinomials

Consider the following: $(X_1, X_2, X_3, X_4) \sim \mathrm{Multinomial} (n,\mathbf{p})$ where $\mathbf{p} = (p_1,p_2,p_3,p_4)$. I would like to find the distribution of $X_1 X_4$, or at least know ...
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1answer
32 views

Definition of standard deviation and $l_2$

If we denote the mean as $\mu$, then the standard deviation is: $$\sigma\equiv\left(\sum_{x\in X}{p(x)(x-\mu)^2}\right)^\frac{1}{2}$$ In other words, $\sigma$ is the average $l_2$ distance from $\mu$. ...
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64 views

How to take into account uncertainty on number of events

Suppose I generate a set of events $X_{i}$ for $i = 1,2 \dots N$ and suppose every event is either a success or a failure, ie. $X_{i} = 0, 1$. If $N$ is fixed, the MLE for the probability of success ...
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1answer
34 views

Applications of statistics to pure mathematics [on hold]

Are there any "applications" of statistical methods to pure mathematics?
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1answer
54 views

A related problem regarding Normal Distribution (Continuous Probability) [on hold]

A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 140 feet and ...
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1answer
53 views

Coin Toss Experiment

I conducted an experiment where I tossed a coin $n=100$ times. I am assuming that the coin flips heads with a probability $p=0.5$. So that the coin is fair with a level of significance of $5%$, I want ...
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4answers
53 views

Binomial distribution, given the number of success, what is the expected total number of trials?

For a random variable that follows binomial distribution, $X|N=n\sim Binomial(n,p)$. What is the expectation of $N$ when we know the value of the random variable but don't know the total? ie. What is ...
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0answers
13 views

Statistics Sampling Type

My question is on Q7. I can't seem to figure this one out. I thought it was a random statified cluster sample because it is breaking down the schools into subsections and then pulling 3 homerooms ...
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13 views

Uncertainty of calibrated Bevel Protractor

I have one problem regarding to ''Uncertainty of Universal Bevel Protractor''. I want to verify a External certificate which is calibrated from outside Lab. Can 'Calibration Uncertainty' be bigger ...
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12 views

Proving convergence of inverse covariance matrix (aka precision matrix)

Let $\Sigma$ be the population covariance matrix and $\hat{\Sigma}$ be the sample covariance matrix. It is well known that $\hat{\Sigma} \rightarrow \Sigma$ in the large sample limit. I have also ...
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1answer
39 views

Continuous distribution and independence [on hold]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
2
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0answers
64 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
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55 views

Probability and continuous distributions

Suppose that the daily consumption of pepsi in ounces is normally distributed with normal(13, 4) in ounces. The daily amount consumed is independent of other days except adjacent days where the ...
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1answer
25 views

Relationship between chi-square goodness-of-fit testing and chi-square distribution?

Anyone care to explain the relationship between a Chi-square goodness of fit test and the Chi-square distribution? One has the expected value in the denominator and the other has the variance in the ...
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1answer
19 views

How are Chi Square probabilities calculated?

What steps would one follow to calculate the values in a Chi Square probability table such as https://people.richland.edu/james/lecture/m170/tbl-chi.html? Say you had 15 degrees of freedom and wanted ...
2
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1answer
38 views

Probability of histogram bars

Say I collect data that follows a Normal distribution $f(z)$ in a histogram with bins of width $w$. I want to calculate the probability that the number of hits $N_i > N_j$. My naive approach would ...
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Concentration inequalities for product of gaussians

Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ gaussian random variables with zero means and equal ...