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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3
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“Mastermind”-esque safe opening problem.

I read this interview question for a trading job and it seems quite difficult. What is the technique to solving it? You have a safe with six digits and a light. You can input a code, if you have ...
2
votes
2answers
27 views

Statistics question on basil bush random variable

The height, $H$, in meters of a basil bush is a random variable with the probability density function $f_{_H}(t)=e^t,\;0\leq t\leq H_0$ such that $H_0$ is the maximal height. $\color{blue}{(1)}$ I ...
4
votes
0answers
47 views

Finding the expected value and variance of ${X^3}$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $\color{blue}{X^3}$ and I know that: cumulative ...
2
votes
0answers
15 views

Estimating the Average and Standard Deviation of a Population based on a Sample with Missing Data with Known Ranks

I need a way to shows me how the parameters of PDF, log-normal in this case, can be estimated based on a set with missing data points at the tail end of a sample. For example, Consider we had 20 ...
0
votes
1answer
28 views

sufficient conditions for a stochastic process to be wide sense stationary

From the page Stationary process, I have the following definition: WSS random processes only require that 1st moment and autocovariance do not vary with respect to time and from the page ...
0
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2answers
27 views

The random variable $ Z = 1-F(X)$

I will formulate the theorem (with no proof) if $X \in \mathbb{R}$ is a random variable with continuous distribution function $F$ then the random variable $Z = 1-F(X)$ has a uniform distribution on ...
1
vote
1answer
40 views

Finding the expected value and variance of $X$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $X$ and I know that: cumulative distribution function: ...
0
votes
1answer
21 views

Finding the probability density function and the distribute accumulate function

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the probability density function and the cumulative distribution function of $X$ My attempt: ...
-3
votes
0answers
17 views

Minimum sample size [on hold]

I have a survey of $n$ distinct opinions and their outcomes rounded to percentages. How can I compute the minimum number of sample size? For example IF resuls are 0.33, 0.33, 0.33 then the minimum ...
-1
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0answers
26 views

different combinations of numbers [on hold]

Can anyone help me pick 1,000 combinations of six numbers from two separate pools of numbers - five different numbers from 1 to 75 and one number from 1 to 15?
2
votes
3answers
46 views

compute temporal average of $\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)$

assuming that $\Phi$ is uniformly distributed over $(0,2\pi)$ compute: $$E[\sin(\omega_0t+\Phi)\sin(\omega_0t+\omega_0\tau+\Phi)]$$ I have solved the problem as continues: $$\begin{align} ...
0
votes
0answers
18 views

Finding upper critical value with Chebyshev's inequality

Consider $X$ is a Poisson random variable with distribution $X$~$Pois(\theta)$. I define the mean in my hypothesis as $\lambda$ and nominal significance level $\alpha$. Null hypothesis $H_0 : ...
1
vote
1answer
39 views

Why this process is nonergodic?

I am studying a tutorial on stochastic processes and there's an example in it which I don't understand anything of it. First of all there is this criterion for a mean-ergodic random process: For ...
1
vote
2answers
24 views

maximum likelihood estimator for theta [on hold]

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 ...
0
votes
0answers
16 views

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 ...
0
votes
2answers
11 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 $X_1, X_3$ ...
2
votes
0answers
35 views

Covariance of 1-D random process is $n\times n$!!!!

I'm reading a tutorial on stochastic processes. There is an example in the tutorial as follows: General Moving Average random process given as $X[n]=\frac{(U[n]+U[n-1])}{2}$ where $E[U[n]]=\mu$ ...
0
votes
1answer
33 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 ...
2
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3answers
45 views

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 ...
2
votes
1answer
30 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 ...
0
votes
1answer
20 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 ...
1
vote
1answer
18 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} ...
0
votes
0answers
7 views

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 ...
0
votes
0answers
41 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 ...
1
vote
1answer
18 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 ...
0
votes
1answer
14 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 ...
5
votes
1answer
43 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 ...
1
vote
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 ...
0
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0answers
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 ...
-1
votes
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 ...
2
votes
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 ...
0
votes
2answers
33 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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votes
0answers
21 views

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?
0
votes
1answer
30 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?
1
vote
0answers
24 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 ...
2
votes
1answer
38 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 $$ ...
2
votes
1answer
60 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 ...
0
votes
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 ...
0
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0answers
26 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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0answers
13 views

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 ...
0
votes
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 ...
0
votes
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
26 views

Are the following partial derivatives calculated from log-likelihood correct?

I am trying to find the double derivatives of the unknowns which form the Fisher Information Matrix for a univariate linear linear Moving Average model. Fisher Information Matrix which is evaluated at ...
1
vote
1answer
42 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 ...
1
vote
1answer
19 views

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!
2
votes
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 ...
1
vote
1answer
34 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 $\mu$. ...
4
votes
2answers
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 ...
1
vote
1answer
34 views

Applications of statistics to pure mathematics [on hold]

Are there any "applications" of statistical methods to pure mathematics?
-1
votes
1answer
56 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 ...