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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Two-parameters Wiener process

Two-parameters Wiener process $W(r, u), r \in [0, 1], u \in [0,1]$ is a stochastic process with a covariate kernel $\mathbb{E}\left[W(r_1, u_1) W(r_2, u_2)\right] = \min(r_1, r_2) \min(u_1, u_2)$. ...
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20 views

Finding the CDF of a piecewise PDF

So for my statistics class I am taking this semester we've been working on continuous random variables and we have one question that the teacher did not cover at all nor his notes, and it has to deal ...
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9 views

find the Expected Value: (1/n^2)E[(ln(x1)+…+ln(xn))^2]

I need to find the Expected Value: (1/n2 )E[(ln(x1)+...+ln(xn))2 ] and the only information I get is that E[ln(xi)] = a, E[ln(xi)2 ] = 2a2 where i = 1,2,...,n. The problem I'm having is with the ...
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GLM for normal distribution

$Y_i$~N$\left(\mu_{i,}\sigma^2\right)\space \mu^2=\alpha+\log \left(\beta_0+\beta_1x_i\right)\space \alpha\space is\space unkown$ how is this proved to be a Genralized Linear Model? My assumption ...
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19 views

Inverse Square Root Of Matrix

So let's say a matrix is A. Then how do you find A^-1/2? It seems to be different from finding the inverse of A. Could someone provide a simple example as ...
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49 views

Exponential moments and integrability

Given a probability distribution $P(x)$ for which the moments $\int x^rdP(x)$ and the exponential moment $\int e^{Rx}dP(x)$ exist for some positive integers $r$ and $R$, than is it true that ...
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15 views

Paired and unpaired data-Statistics/Hypothesis testing

I'm getting a bit confused about paired and unpaired data, for example in this question I don't understand how this data is paired. If it was the same steel pipes that were left uncoated first in ...
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11 views

Statistical tolerancing of 2 independent, normally distributed, variables. What is the reliability probability?

The image below is of a solid, machined part. Ignore everything aside from the red, that is where a slot has been cut to fit a block. With the assigned tolerances what is the probability that a ...
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21 views

Statistics - 95% confidence level?

You are to conduct a sample survey to determine the mean family income in a rural area. The question is how many families should be sampled? In a pilot sample of $10$ families, the standard ...
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1answer
30 views

Normal Distribution: Covariance of sample mean and sample

$X_1,\ldots,\overline{X}_N$ is a random sample of $N(\mu,\sigma^2)$ observations. Let $\overline{X}_N$ denote the sample mean. What is $\operatorname{cov}(X_1, \overline{X}_N)$? Since the mean of ...
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Generating simulated data [closed]

I have 1 day simulated data for a shop in excel sheet, how can i generate simulated data for hundred days in excel? I have added image for more information about what kind of data i have.
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8 views

Stability of boostrap confidence intervals

As a word of background, I want to show that certain result is stable when averaging over a large number of simulations, but could be just a lucky draw with a small number of simulations. I have a ...
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15 views

Systems of equations that are over and under-determined at the same time

I know what an over and under-determined system is. What I am not sure is how to show both. I am asked to come up with a matrix that is both an under-determined and over-determined system which is the ...
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16 views

Conditional expected waiting time for a bus to arrive given number of people waiting

Suppose there are $N$ people and each person arrives at the bus stop after a waiting time distributed i.i.d. as an exponential distribution with parameter $\lambda$. Suppose further that the bus ...
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24 views

normal approximation of binomial distribution

a school buys 60% of its light bulbs from supplier A and 40% from supplier B. the light bulbs from both suppliers look identical but light bulbs from supplier A have exponentially distributed ...
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1answer
23 views

How can I calculate Index of Coincidence of Vigenère cipher?

I have computed the letter frequency of the cipher text. However, I don't know how to apply Friedman Test to Vigenère cipher. I couldn't calculate the Index of Coincidence. Is anyone can help ? ...
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2answers
40 views

Over-determined and Under-determined systems

How do I show that a system is both over-determined and under-determined? I am supposed to come up with a matrix that satisfies both but am not really sure I understand what types of equations would ...
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1answer
48 views
+100

Integrate two sets of Data and check similarity

I have the following CSV Data I used Excel Charts to plot the Data I want to compare these Data with other Data, in another word i want to comapre 2 curves and check the similarity between them in ...
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27 views

How can we terminate in a finite amount of time?

What's the difference between: 1.Always terminates in a finite amount of time 2.Terminates in a finite amount of time with probability 1 Extension: Are there any more ways we could add to the list? ...
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17 views

Scratch card odds

I'm trying to work out how to set up a scratch card system with a preset probability. Lets say I have types going from 1-8 which corresponds to your chance of winning. So a type 8 card is the ...
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2answers
26 views

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y?

If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y ?
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24 views

How do I compute this probability? [closed]

Suppose that every morning after the 8:30 bus leaves a bus stop, two passengers arrive. The first passenger arrives $X_1$ minutes after 8:30, and the second $X_2$ minutes after the first. If the $X_j$ ...
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1answer
19 views

Statistics - Mean value and net revenue

From the following book:Probability and Statistics for Engineering and Science
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35 views

Identifying Break-Points in Binomial Responses (Alternative to Multiple Chi-Squared Tests?)

So I have a research question whereby I have subjects' age and a yes or no response to a question. Ages range from 45-90 approx. I am currently looking at whether age has an affect on these yes or no ...
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how to find $X^2_{0.95,14}$, $P(X^2_{9} \geq y) = 0.99$ in a TI-84 calculator?

Can someone please show the steps in how to find $X^2_{0.95,14}$ in a TI-84 calculator? My professor does not want us to use a statistics table, but to know how to use the calculator in quizzes and ...
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Minimizing cleansing cost [closed]

Please find attached the images of questions. Help me out if you can. THe question is simple yet i am clueless.
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statistics dice problem

If $5$ fair dice are thrown at the same time, how do you find the probability that there are three $1$'s and two $2$'s? The answer says its $5C2 \cdot (1/6)^5$ but I don't understand why.
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37 views

chose the best path for estimation

I have a Cartesian grid (100x100) in which some of the points are known (30 out of 10,000) and the rest are unknown. I want to use the known points and estimate the other cells. Is there any ...
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21 views

Gaussian distribution finite population with unknown cardinality

I have taken a sample population of a population with unknown size. The sample size is 54 trades. The sample mean is 2.1% (1.021) return per trade. The sample standard deviation is 0.01. 100% of ...
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1answer
23 views

monte carlo simulation: multiple variables

I am reading about the merton model in finance. It depends on multiple distributions. I want to use monte carlo simulation but I've a little question about this. If we have a statistical model that ...
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1answer
16 views

standard error for the parameters of a linear regression model

Given a linear model $\mathbf{y} = \beta \mathbf{X} + \epsilon$, it is well known that the estimate for $\beta$ that gives the minimum residual sum of squares (RSS) is given by $\hat{\beta} = ...
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references on social physics [closed]

I've been very curious about social physics since the moment I read the book "Big data and social physics: The lessons from a new science" written by Alex Pentland. I also know that there are many ...
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Scatter plot predict/forecast values based on historic values

I do not know if this is the correct forum, as you guys are good at math I'll give it a shot here. Now I've been thinking about this for awhile and have not found out any statistical/mathematical ...
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1answer
27 views

Probability, binomial?

I'm working on a programming assignment and am completely stuck on a problem involving statistics :/. In my problem, I am dealing with a hypothetical situation where I am walking outdoors and am ...
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How to drop variables in statistical classification analysis?

Given a set of data with variables and a training set, we can proceed classification analysis using Mahalanobis distance etc.(discriminant methods) But how do we know whether all these given ...
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51 views

$E(X^2)=E(X)=1$. Find $E(X^{100}).$

$X$ is a random variable such that $E(X^2)=E(X)=1$. Find $E(X^{100}).$ My attempt: Assuming $X$ is discrete, we have $\sum x_i\mathbb P(X=x_i) = \sum x_i^2\mathbb P(X=x_i) = 1.$ We have something ...
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28 views

Percentiles and Salary Surveys

What formula would I use (using Excel, preferably) to determine the 'average' 10th percentile rank value if I have four different and yet related 10th percentile values? Can I just average the values ...
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What is the name (if any) of this statistical analysis concept?

I have a list of random numbers. I sort them, and designate each one as belonging to a tier of some sort (quartiles, percentiles, or even arbitrary cutoff points by proportion) based on the ...
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How do I evaluate $\mathbb E(X\log(X))$ if $X$ has a binomial distribution, for large $n$ values?

$X\sim\mathcal {Bin}(n,p)$ I want to evaluate $\sum\limits_{x=0}^n {^n\mathrm C_x} p^x(1-p)^{n-x}x\log(x)$. Is there any way to avoid the sum because my $n$ can be very large (around $10^6$)?
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42 views

How to find conditional expectation $\mathbb E(X|X<M)$

Consider a random variable $X$ following the so-called folded normal distribution. That is, $X$ has density function $$ f_X(x) = \sqrt{\frac{2}{\pi\tau}}e^{-\frac{x^2}{2\tau}}, x>0. $$ ...
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Power spectral density of the system output

$w(t)$ and $z(t)$: two stationary random processes $z(t) = Pw(t)$. $P$: a stable, LTI system. How to show: $$ S_z(jw) = P(jw)S_w(jw)P(jw)^*$$ $S_z(jw)$ is the power spectral density of ...
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How to estimate parameters in trigram?

A popular method of computing trigram in NLP is linear interpolation: The question is how to estimate the three linear interpolation parameters to maximzie the following expression? Any form of ...
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Generalized Linear Model

$y_i$~$\space poisson\space with\space \ mean \space \mu_{i\space }where\space \mu_{i\space }=n_ie^{\beta\ x_i}\space and\space n_{i\space }is\space kown$ I have an exam coming up shortly where ...
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How to get the ratio distribution of $Z=Y/X$ in this way? [closed]

I know how to do this: $$F(z)=\Pr(Z ≤z)=\Pr(Y/X ≤ z)=\Pr(Y/X ≤ z , X < 0) + \Pr(Y/X ≤ z , X ≥ 0)$$ But, I don't know how to do this: $$F(z)=\Pr(Z ≤z)=\Pr(Y/X ≤ z)=\Pr(Y/X ≤ z , Y < 0) + \Pr(Y/X ...
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why do I get imaginary number using Discrete Fourier transform?

I am looking for an approximation for Poisson binomial distribution: The Poisson binomial distribution is the discrete probability distribution of a sum of n independent Bernoulli trials. you can ...
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Hypothesis Question

Consider the following hypothesis: $H_{0}:\mu\leq3000$ vs $H_{a}:\mu>3000$ A sample size $n$ must be decided so the risk of a type 1 error is at most 1%, and also so that if the value of ...
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geometric standard deviation

The geometric mean is like the arithmetic mean on a log scale. with the arithmetic mean it is often useful to find the standard deviation. Can the same sort of thing be done to create a geometric ...
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optima; value of a function

Suppose we have the following function $$Err(f) = \frac{1}{2}E|Y-f(X)| = P(Y=1,f(X)=-1) + P(Y=-1,f(X)=1),$$ where $Y, f(X) \in \{-1, 1\}$. How can find the optimal value of the above function, Err? I ...
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How do I calculate PERT stddev if I use custom weight factors? [migrated]

According to (for example) http://tynerblain.com/blog/2006/04/13/foundation-series-basic-pert-estimate-tutorial/, the PERT mean estimate is $$meanA=optimistic+(4*likely)+pessimistic)/6$$ I ...
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How to find the cdf of $U=y-x$

The random variables x and y have the probability density functions $f_{X,Y}(x,y)=4e^{-2(x+y)}$. How do I find the probability density functions of $U=y-x I have tried solve by convolution theorem ...