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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How the formula for Pearson correlation coefficient is derived?

Recently I have learned using the sample correlation formula to find out the linearity of paired data set (x,y). I was told the properties when the coefficient r is equal to 1/0/-1 but I was never ...
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1answer
17 views

Unpacking the definition of a likelihood function

I am learning stats. My book says let $X_{i}, ..., X_{n}$ be IID with PDF $f(x;\theta)$. It then defines a likelihood function as $$\displaystyle L_{n}(\theta) = \prod_{i=1}^nf(x;\theta)$$ I think ...
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9 views

Problems understanding Autocorrelation/Autokovariance

I am having some trouble understanding the concept of autokovariance/autokorrelation with a timelag l. From how i understand it, it is the kovariance/korellation a series has, with a timelagged ...
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20 views

At which value is the density in two poisson-processes the same?

Two poisson processes, $X$ with $\lambda=3$ events pr. hour and $Y$ with $\lambda=5$ events pr. hour. Let $Y$ be the waiting time. At what value, $y$, in minutes and seconds, is the density, ...
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18 views

Gaussian mixture model - mode finding

I'm doing some research, and I stuck at finding mode for gaussian mixture model. I have set of $3D$ points. For each pair of points, I need to calculate following: $\pi_{ij}$ - probability that ...
0
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10 views

Gap test for RNGs

Gap test is discussed here gap test for random numbers , if we have probabilities how we can find the expected number of gaps in order to compare with counted number of gaps after running random ...
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7 views

Completeness - Normal iid sample with the variance known

How to prove that the family of distributions for a normal iid sample with known variance is complete without using the result that this normal family is a member of the exponencial family?
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22 views

Statistical distribution methods

I'm coding a program to classify a stock price movement in a given timeframe. I have around 20k data points and it ranges from 0 to around 300, with the average being 10,8. I want to use a genetic ...
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1answer
16 views

Excpected value of unbiased estimator.

Let $ \Theta$ is any parameter of any statistic such that $\Theta $ has an unbiased, normally distributed estimator $\overline \Theta$. Now, it is written in the book, that: $$E \overline \Theta = ...
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1answer
33 views

Mean and standard derivation

A set of numbers consists of one's and three's. Find the mean and standard derivation if there are 23 one's and 17 three's. What's the meaning of this question ? I know the formula for mean and s.d. ...
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1answer
27 views

Finding the mode of the negative binomial distribution

The negative binomial distribution is as follows: $f_X(k)=\binom{k-1}{n-1}p^n(1-p)^{k-n}.$ To find its mode, we want to find the $k$ with the highest probability. So we want to find $P(X=k-1)\leq ...
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3answers
36 views

What is the relationship betweeen a pdf and cdf?

I am learning stats. On page 20, my book, All of Statistics 1e, defines a CDF as function that maps x to the probability that a random variable, X, is less than x. $F_{x}(x) = P(X\leq x)$ On page ...
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16 views

Probability with a table

a) P(randomly picking someone going to the concert)= I have .406 but its wrong b) P(randomly picking someone who has kids)= I have .54 but it might be wrong c) P(randomly picking someone who does ...
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1answer
11 views

What does this author mean by “limiting frequency” (in the context of frequentist statistics)

I am reading All of Statistics by Larry Wasserman. On page 176, he says that Bayesan statistics is different than frequentist statistics, in part, because Bayeseans believe that "probability describes ...
3
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0answers
15 views

Behavior of MGF of Quadratic Combination of Dependent Multivariate Gaussians

Sorry if the formatting is poor, this is my first time asking a question. I'm investigating how squared gaussians behave, using the techniques provided here, which are giving me inconsistent results. ...
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1answer
15 views

Random variables vs sample of mesurement

i would like to know why a sample of measurement in Statistics is being represented through random variables even though it's about only one quality of measurement, say the weight of individuals of a ...
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1answer
12 views

Calculating Margin Of Error

I'm having trouble calculating the margin of error for this problem. "What is the overall margin of error of the survey, at 95% confidence, if it is based on a simple random sample of 1111 voters?" ...
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2answers
32 views

Finding the mode of a distribution

I've been trying to get a better understanding of distributions. So far I understand how we get the formulas for mean and variance (by looking at the derivative of the moment generating function). ...
2
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1answer
32 views

Function with Multiple Periods

Basically I'm trying to fit some data with seasonal effects to a periodic function, and the problem I'm running into is that the local minima usually occur around April, and the local maxima usually ...
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26 views

Percentage and stats

In Biology class you have $4$ tests, each worth $15\%$ of your final grade, and a final exam worth $40\%$. If your test scores are $80$,$80$,$95$ and $65$; and your final exam score is $85$, what is ...
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13 views

Is the SemiCovariance matrix positive definite?

I am trying to proof whether the SemiCovariance matrix defined in equation (5) from the following paper http://web.iese.edu/jestrada/PDF/Research/Refereed/MSO.pdf is positive definite or not. As a ...
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votes
2answers
21 views

Percentiles and percentage [on hold]

My test grade of $83$ falls in the $74th$ percentile of all grades. About what percentage of students did I score higher than? I thought it was $61.42\%$ but that wasn't right. I thought it was ...
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8 views

hypothesis test type 1 and 2 error

Supporters claim that a new windmill can generate an average of at least 800 kilowatts of power per day. Daily power generation for the windmill is assumed to be normally distributed with a standard ...
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17 views

multivariate interval estimation

I have several samples of probabilistic vectors, i.e, each sample is of the form $(x_1, \cdots, x_n)$ such that $\sum_{i=1}^n x_i\leq 1$ (they are sub-probabilistic vectors), how can I obtain a ...
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17 views

How do i find the percent of packages that will be delivered in less than 16 hours

Couruer service has found that its delivery time is normally distrbuted with a mean of 14 hours and a standard deviation of 3 hours. what percent of packages will be delivered in less than 16 hours?
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12 views

Additive and multiplicative model

I'm new in study of survival analysis. There are some basic-questions that I wanna ask and it will be great if somebody could help me to aswer these following questions: What the differences between ...
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1answer
16 views

p value and hypothesis testing

In a random sample of 99 games the home team won 57 games. Test the null hypothesis That home team wins one half of all games against the alternative that the home team wins a majority games. How can ...
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1answer
30 views

Simultaneous density function of two continuous variables, X and Y.

I'm having issues with calculating the simultaneous density function of two continuous variables, X and Y. I took a screenshot of the information: How should I start? I know that if the two ...
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0answers
9 views

How to combine 2 different variables (time&cost) into a fitness function?

We are working on a simulation of uber taxi drivers, and to measure the effectiveness of additional functionality (like picking up extra passengers along the way) we would need to create a fitness ...
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28 views

Hoeffding’s inequality extension

In Hoeffding’s inequality we assume that the random variables $X_i$ ,$i=1,..,n$ are i.i.d. and bounded . Is there any extension to Hoeffding’s inequality for the case that $X_i$ are identically ...
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1answer
32 views

Steve Nash’s expected value from his one-and-one free throw situation is 1.72 points. What is his free-throw percentage?

The one-on-one free throw situation works like this - for the first throw, if you make it, you get to do it again. If you miss, you don't get another chance. If you make it the second time, you get ...
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15 views

Hypothesis test, p value problem self study

In a random sample ,p=57/99 Hypothesis test H0 : P = 1/2 H1 : P > 1/2 I want to test the H0 and want to find the p value and threshold of significance level. I am trying to solve this problem. I ...
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3answers
26 views

How to denote an average of data satisfying a given condition

I have to write an expression for the following. I have values, one fpr each of the last 5 months: ...
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14 views

bias reduction when the bias depends on the true parameter

Let's say we estimate a parameter, $\theta$, by $\hat{\theta}$. For this estimator we have the following property that $$\hat{\theta}\to_{p}\theta+f(\theta)$$ where $\to_{p}$ denotes convergence in ...
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1answer
34 views

How to use Bayes' Theorem to forecast crude oil prices? [on hold]

So I'm new with Bayes' Theorem, but I have to use it to predict future crude oil prices. I have been looking at examples online, but I got very confused by it. I can understand the simple examples ...
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36 views

Urn problem with replacement where the amount of balls drawn is random [on hold]

Suppose we have an urn with 52 balls, all are numbered from 1 to 52. We draw twice from the urn with replacement. The amount of balls drawn every turn is random with the probabilities p_0,p_1...p_8 ...
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15 views

What is a easy way to draw from distribution $f(x|u) \propto x^{\alpha-1}I(x<u)$?

We know $u$ and want to draw $x$ from the conditional density $f(x|u) \propto x^{\alpha-1}I(x<u), \alpha>0$. One way is that first draw $r$ from uniform(0,1), and then set $x=u ...
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14 views

What statistical methods can I use to compare the differences between the two time series?

I conducted gas emission measurements from two buildings for a year (one is the building without treatment; the other is the one with treatment). They are two time series data. I plotted the data and ...
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3answers
76 views

Suppose a city with Three type of coins ?!

in a city we have tree type 1 dollar, 2 dollar, 3 dollar of coins. we want to pay for a 20 dollar product. how many ways we can pay for a 20 dollar product, if the seller has no money and number of 1 ...
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1answer
34 views

Marginal distribution of uniform distribution conditioned on poission?? [on hold]

Let $N$ be a Poisson distribution with parameter $\lambda$. Conditioned on $N$, let $X$ be uniformly distributed. What is the marginal distribution of $X$? ( this is one of my final exam problems ...
0
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1answer
33 views

Geometric Random Variables

I have a question that involves a certain criteria of a random variable as shown: The random variable $X$ has the distribution $Geo(0.2)$ and I would love it if someone could explain what the ...
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0answers
57 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or more? [on hold]

I have made a probability game, where you have to pull out any 5 cards without looking (from a deck of 52 cards), and if all five cards add up to 40 or more, they player pulling the 5 cards from the ...
0
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3answers
30 views

Proof variance of Geometric Distribution

I have a Geometric Distribution, where the stochastic variable $X$ represents the number of failures before the first success. The distribution function is $P(X=x) = q^x p$ for $x=0,1,2,\ldots$ and ...
2
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1answer
60 views

What's the probability that the first four children born are boys and the last two children born are girls?

I'm having some problems with determining how to calculate a question about the gender proportion in newborns in some random family. A family consists of 6 children. The probability of a boy being ...
0
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1answer
35 views

Pareto distribution

So I'm given a Pareto distribution with parameters $\alpha >0$ and $k>0$ which is the form of \begin{equation*} f(x) = \frac{\alpha k^\alpha}{x^{\alpha+1}},~x > k. \end{equation*} I found ...
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25 views

Combining Sample Means and Variances

Let us assume we have several samples of unknown size but known mean value $\mu_{i}(x)$ and known variance $\sigma_{i}^{2}$. Now we want to calculate the mean value and variance for the total ...
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21 views

Solving an integral to solve a statistical problem

In solving a statistical problem, I want to know $$\mathbb{P}( (Y_2 \leq q_\alpha | Y_1 \geq q_\alpha) $$ where $Y_1 = X_1 + Z$ and $Y_2 = X_2 + Z$ and $Z \sim N(0,\sigma_z^2), X_i \sim N(0, ...
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13 views

Does Weighted avg of Weighted avg makes sense..?

Scenario is like we have a city in which there are many schools. All of them conducted a mathematics exam and it depends on the school for each class how much marks the exam would be. Suppose school ...
3
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3answers
31 views

Symmetry in Probability Around a Particular Phenomenon in Time?

This has been hurting my brain substantially, recently. I'm not sure if I'm failing to make connections or if I see connections but am weary of their relevance. In my text the author claims that ...
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7 views

What is Bootstrapping in statistics? How can I use it to determine error in the mean, variance, kurtosis and skewness of a data set?

From what I understood from searching randomly is that it has something to do with resampling. What does this resampling mean? Is it selecting random data from a distribution or is it getting data ...