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.

learn more… | top users | synonyms

3
votes
2answers
4k views

Stochastic Process Examples

I was wondering if people could give me examples of how stochastic processes are seen and used in research in real life.
1
vote
2answers
527 views

Bivariate Normal Distribution: Finding the joint distribution of functions of random variables

I need your help with this problem: Suppose $(X, Y)'$ follows a Bivariate Normal Distribution with parameters $μ_1 ,μ_2, σ_1^2, σ_2^2$, and $ρ$. Let $U = X + Y$ and $V = X - Y$. Considering that $X$ ...
0
votes
1answer
60 views

How to get calculate ratio and compare between two

I am making report, I need help finding who performed best. I have 3 workers, and they are assigned number of work/task to do. but but tasked assigned to each of them are not same numbers. How can ...
1
vote
1answer
323 views

Numerical calculation of fisher information

I am trying to obtain numerically the fisher information. Given a likelihood function $$ f(X,\theta),$$ with $X \in [0,1]$. The fisher information is given by $$ ...
-2
votes
1answer
297 views

finding UMVUE of parameter $\displaystyle 3^{-\theta}$ [closed]

suppose $X_1,X_2,\ldots,X_n$ is a random sample of $Beta(\theta+1,1)$. how can find UMVUE of parameter $\displaystyle 3^{-\theta}$
-1
votes
1answer
1k views

Finding the ACF of AR(1) process

For an AR(1) process: $X_{t} = \phi X_{t-1} + w_{t}$ with $w_{t} \sim N(0,\sigma^{2})$ How do you derive the ACF of the process? Since $E[X_{t}] = 0$, would you just calculate $cov(\phi X_{t-1} + ...
10
votes
3answers
2k views

Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
2
votes
1answer
162 views

Bernoulli distribution: expectation problem with independent random variables

We have $X_1,...,X_n$ as $n$ independent random variables under the Bernoulli distribution i.e.: $$P(X_i=1)=p$$ $$P(X_i=0)=1-p$$ where, $p$ is an unknown parameter. The distribution $Y=\sum_1^n X_i$ ...
0
votes
1answer
106 views

Type II error - Hypotheses

A study of MBA graduates by University of Oregon Survey $1999$ revealed that MBA graduates have several expectations of prospective employers beyond their base pay. In particular, according tro the ...
0
votes
1answer
21 views

calculating quantiles of certain divisions

Is there any formula or other way to calculate: The $p$ quantile of $X\sim\exp(1)$ (or generally $x~\exp(λ)$) The $p$ quantile of $X\sim\text{uni}[a,b]$ thanks in advance.
0
votes
1answer
269 views

Find standard deviation given standard deviation

How would I find the standard deviation of a value, V that is the average of other values, say heights of people, given that I have the standard deviation of the heights? I'm looking to improve my ...
1
vote
2answers
580 views

Matlab, finding the variance given a probability distribution

How would I numerically find the variance given a probability distribution in MATLAB? I understand(I think) how to do it symbolically, but is there any way to be confirm my results? I am given the ...
0
votes
1answer
94 views

Fitting a model in R, Least squares estimates

$ \Bbb E$(grades) = $\bigg(\alpha$, teacher='Female', and, $\bigg( \alpha+\beta $, teacher='Male' given that I have loaded up the required data, how should I go about fitting this model in R? Then ...
0
votes
1answer
69 views

p-Value for regression on the whole population

I would like to know what is the meaning of the p-Value statistic when I make a linear regression with the full population (and not a small sample of it). Thank you
0
votes
2answers
61 views

Expected value given distribution

What would be the variance of a random var. $Z$ with distribution $\mathbb{P}(Z=n)=2^{-n}$ over all positive integers? I am clueless. I know $\mathbb{E}(Z)$ would be $\sum_{n=1}^\infty n 2^{-n}$. At ...
2
votes
2answers
46 views

finding $P(X_1>X_2\mid X_1>X_3)$ in Exponential distribution

suppose $X_1,X_2,X_3$ are independent random variables with exponential distribution and means $\frac{1}{\lambda1},\frac{1}{\lambda2},\frac{1}{\lambda3}$ , how can find $P(X_1>X_2\mid X_1>X_3)$
1
vote
0answers
192 views

finding $cov(X,Y)$ [duplicate]

suppose $X\sim N(0,1)$ , $Y = \left\{ \begin{array}{ll} X, & \hbox{if } |X|\geq a, \\ -X, & \hbox{if }|X|< a. \end{array} \right. $ how can find $cov(X,Y)$. $\phi,\Phi$ are density ...
1
vote
1answer
225 views

Brownian Motion and the Functional CLT

Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
1
vote
2answers
220 views

One question regarding r.v independence

I just encounter independence in a Statistics course, I get stuck in this question for a long time..any help will be extremely appreciated. First one is, if $X_1, X_2, X_3...X_k$ (finitely many) are ...
2
votes
1answer
77 views

Proof related to Chebychev's inequality

I need to prove that in a set of $N$ data $x_1, x_2, \ldots, x_n$, for all $i$ between 1 and $N$, we have $$\mu-\sigma \sqrt N \leq x_i \leq \mu+\sigma \sqrt N$$ where $\mu$ is the average and ...
1
vote
3answers
57 views

Having trouble with expected values

What would be the expected value of a random variable with distribution $\displaystyle \frac{1}{k^2-k}$? I'm basically stuck. $k \in [2, \infty) \cap \mathbb N$.
0
votes
1answer
33 views

Simple statistics - Average and global average

There are two groups of English: intermediate and advanced. The average girls in the intermediate group is higher than that of the boys in the same group; The average girls in the ...
0
votes
1answer
222 views

Pearson Correlation Coefficient Interpretation

Let $X=(1,2,3,...,20)$. Suppose that $Y=(y_1,y_2,...,y_{20})$ with $y_i=x_i^2$ and $Z=(z_1,z_2,...,z_{20})$ with $z_i=e^{x_i}$. Pearson correlation coefficient is defined by formula \begin{equation} ...
1
vote
1answer
104 views

Combining independent probabilities: ghosts in a haunted house

Despite taking several statistics courses in college years ago, I've never had a mind for probabilities, so please excuse what may be a very basic question. To make this fun, let's say you are in a ...
1
vote
2answers
69 views

Interpretaton of confidence interval

I have read two books that explicitly state that the $(1-\alpha)$% confidence interval should be interpreted as: If you construct 100 such confidence intervals, $\alpha$ of them are expected to not ...
0
votes
0answers
67 views

Prove that $l(\tau\sigma n^{(-1/2)})$ tends to $\chi^2_1(\tau^2)$

Suppose that $X_1, \ldots ,X_n$ are i.i.d. sample with $E(X_1) = 0$ and $E(X_1^2) = \sigma^2 < \infty$. How can we prove that $l(\tau\sigma n^{-1/2})$ tends to $\chi^2_1(\tau^2)$? Here $l(·)$ is ...
1
vote
3answers
113 views

Question on probability

there are two bowls with black olives in one and green in the other. A boy takes 20 green olives and puts in the black olive bowl, mixes the black olive bowl, takes 20 olives and puts it in the green ...
1
vote
1answer
100 views

How to prove the earning decomposition of 2 people in mediocristan and extremistan?

In his book The Black Swan (chapter 15, section The Mandelbrotian), Nassim Nicholas Tayeb says that if the sum of the earnings of 2 people is 1 million, the most probable decomposition in Mediocristan ...
-1
votes
1answer
136 views

Inverse transform sampling

I know the basic idea is to generate a random number from $U(0,1)$, find the inverse cumulative distribution function $F^{-1}$ and then take $x = F^{-1}(U)$. If you were plot a histogram of say 1000 ...
0
votes
1answer
149 views

How to solve multi-variate linear regression analytically?

We have $n$ variables $x_n$ and one stochastic function $y$ of these variables. We assume that function $y$ depends on variables in the following way: $y = c + \sum_{i=1}^n k_i x_i + \varepsilon_i$, ...
2
votes
1answer
121 views

Log-likelihood for multinominal normal distribution

Given $n$ jointly-normal random variables $X_1, X_2, \dots, X_n$, with $$\mu_i=\mu\forall i \in\mathbb{N}^+$$ $$\sigma_i=\sigma\forall i$$ $$\rho_{i,j}=\rho\forall i,j \mbox{ with } i\neq j$$ what is ...
1
vote
1answer
44 views

Independent random variables considering expression

Having $x, y, z, c \in \mathbb{R}$, is it valid to say: $c \propto g(x, z) h(y, z)$ The context here is to say whether or not the random variables $X$ and $Y$ are independent given the value of $Z$. ...
2
votes
1answer
119 views

Maximum likelihood for $(\mu,\sigma)$ and other related questions

$$f(x)=\frac{1}{2\sigma}\exp\left(\frac{-|x-\mu|}{\sigma}\right)$$ $$\mu\in,\sigma>0$$ When trying to calculate the maximum likelihood for $(\mu,\sigma)$, I got as far as: $\log L(\mu,\sigma)=-n ...
2
votes
1answer
125 views

Bias of Estimator with square root of a sum of squared random variables

Got a distribution of $f_X(x;\theta) = (x/\theta^2) \exp(-x^2/2\theta^2)$ for $x \ge 0$ where the MLE is calculated as $\theta_{MLE} = \sqrt{(\sum_{i=1}^{n}x^2_i)/2n}$ So now need to find if it's ...
1
vote
2answers
41 views

(poisson distro) Why is my expansion of the messy summation for the answer wrong?

There is this problem on the textbook. The derivation of the answer in the solution was not very clear, but this is how the problem goes: Calls arrive at an office, following a Poission distribution, ...
1
vote
1answer
90 views

How to sample point from triangle where vertex is not in origin

This link http://mathworld.wolfram.com/TrianglePointPicking.html gives an overview of how to sample points from either a quadrilateral or triangle given one vertex is at the origin. The standard ...
0
votes
2answers
88 views

probability generating function with die

Given a fair six-sided die. Find the probability generating functions for the number of the throw on which the rth six appears. Hence find the probability that the fifth six occurs on the 20th ...
1
vote
1answer
52 views

Inferential Statistics

need help with the problem below. Thank you!! A national survey showed that Puregold cold cuts were priced, on the average, at $5.20 per pound. Supposed a national survey of 23 retail outlets was ...
0
votes
1answer
203 views

Histogram with Gaussian bell curve

How do I create/calculate the probability density curve in a histogram which is scaled to the frequency axis with ABSOLUTE values (example)? The curve should be based on the calculated average and the ...
2
votes
1answer
109 views

How to calculate $x_t$ from $ARIMA(1,2,0)$ (second difference $AR(1)$ process)?

It sounds so simple but I'v struggled with this problem for a quite long time now. I have come to this: $$X_t = \phi(x_{t-1} - 2x_{t-2} + x_{t-3}) + 2x_{t-1} - x_{t-2} + \epsilon_t$$ it just doesn't ...
1
vote
0answers
500 views

Probability that a normal distribution is greater than two others

Given 3 independent variables with normal distributions, how can I calculate the probability that one of them will be greater than the other two simultaneously? So, how to calculate $P ((A>B) ...
-1
votes
1answer
194 views

Odds of winning a race given the odds of each runner beating each opponent?

Say we have three runners: A, B and C, and we have the probability of each runner beating each individual opponent: A before B: $0.68$ A before C: $0.42$ B before A: $0.32$ B before C: $0.30$ C ...
0
votes
2answers
4k views

Probability in a Card Deck using Combinations

Consider the following experiment. Three cards are drawn from a standard deck of 52 cards, one after the other. Before the second card is drawn, the first card is put back in the deck and the ...
0
votes
1answer
268 views

Help with understanding the proof of Hoeffding's inequality?

I'm doing machine learning and now I'm struggling to understand the proof of Hoeffding's inequality. The proof I'm using for learning can be found here: Hoeffding proof Now for starters what bothers ...
0
votes
1answer
53 views

What are the advanatages of CDFs for RNG over simple random sampling?

If I understand Cumulative Distribution Functions (CDFs) correctly, they can be used for random number generation from a given dataset as follows: Build a CDF that maps data points to an ordinal ...
0
votes
1answer
287 views

Limit of sum of (continuous) uniform distributions

In my stats courses at university, I've been working on transformations of distributions etcetera. However, one particular case has intrigued me for a while: the sum of continuous uniform ...
8
votes
1answer
5k views

Proof that the hypergeometric distribution with large $N$ approaches the binomial distribution.

I have this problem on a textbook that doesn't have a solution. It is: Let $$f(x)=\frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}}\;,$$ and keep $p=\dfrac{r}{N}$ fixed. Prove that $$\lim_{N ...
1
vote
0answers
53 views

Rigorous standardisation of data

I have 5 sets of raw experimental data + 1 control set and would like to know what you think about standardising/normalising each set. I want to compare the 5 sets of experimental data to the control ...
1
vote
0answers
68 views

Differentiation help

I recently got some lecture slides, but needed a little help understanding the maths behind them. (equations) (Working and Answer) Basically, I don't understand how to get from step (4) to (5). ...
2
votes
1answer
141 views

Poisson distribution normal approximation

6.4.18. An experimenter takes a sample of size 1 from the Poisson probability model, pX (k) = e−λλk/k!, k = 0, 1, 2, . . . , and wishes to test H0: λ=6 versus H1:λ<6 by rejecting H0 if k ≤2. (a) ...