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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In statistics, what is the reason that makes the average height replace the median height mathematically?

In statistics, what is the reason that makes the average height replace the median height mathematically? When i searched Google there are only average height available but not median height.
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133 views

Prove that $\int_{0}^1 u^{\alpha_1-1} (1-u)^{\alpha_2-1} \, {\rm d}u =\frac{\Gamma(\alpha_1)\Gamma(\alpha_2)} {\Gamma(\alpha_1+\alpha_2)}$

I have the following equality in a textbook of mine $$\frac{y^{\alpha_1+\alpha_2-1} e^{-y/\beta}}{\Gamma(\alpha_1+\alpha_2) \beta^{\alpha_1+\alpha_2}} \cdot \frac{\Gamma(\alpha_1+\alpha_2)}{\Gamma(\...
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How long will it take me to learn math as a beginner [closed]

am 18 years old. I won't like to say i don't know maths, but i have a very poor foundation in maths. I am very weak in maths, even some 8 year olds are better than me. I can answer about 4 out of 20 ...
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42 views

Is it possible for a multinomial sample to be a single number?

I'm reading the Latent Dirichlet Allocation paper trying to understand it. However I got stuck at the very first part! When they sampled from a multinomial distribution and considered the result to be ...
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Linear regression: degrees of freedom of SST, SSR, and RSS

I'm trying to understand the concept of degrees of freedom in the specific case of the three quantities involved in a linear regression solution, i.e. $SST=SSR+SSE, $ i.e. Total sum of squares = ...
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1k views

Computing a combined average (mean) of values

My textbook on descriptive statistics says that when we have a data set of m members, the mean is calculated like this: x-bar = 1/m (Σfixi) where sigma sums values from 1 to k (we have k groups in ...
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39 views

can any one suggest me a way for normalizing data ?

I want to have my data in range of 0-1 . I tried min-max function, decimal scaling but i need other techniques. My data has variable range from 0 - 35000 .
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Given every horse's chance of winning a race, what is the probability that a specific horse will finish in nth place?

I have been interested in calculating a specific horse's chance of finishing in nth place given every horse's chance of winning in a particular race. i.e. Given the following: ...
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79 views

Convergence of random increment of two bins

If bins A and B are initialized to 1, and then I continually increment one of them by 1 with a probability proportional to their values, would A/B ever converge? To demonstrate ...
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86 views

Showing the Posterior distribution is a Gamma

Assume $X\sim \mathrm{iid}\operatorname{Pareto}(a,b)$, and $b \le \min(X)$, then $$f_n({\bf x}; \theta) = a^n b^{-n} \prod^n_{i=1}\left( \frac{b}{x_i} \right) ^{1 + a} $$ We assume b is known and ...
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2k views

How are the values in the Shapiro Wilk weight table calculated?

http://www.real-statistics.com/statistics-tables/shapiro-wilk-table/ I am trying to build a function that performs a Shapiro-Wilk test on a sample population, but I cannot figure out how the a values ...
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1k views

Shapiro–Wilk test calculation in plain terms

Could someone please explain how to perform a Shapiro–Wilk test using an array of values (e.g. (1,2,3,4,5)? I've been searching on Google, but almost all of the results show how to calculate this ...
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2answers
314 views

Derivation $X^2$-pdf (Chi square) for $k$ degrees of freedom

Could someone explain the steps of the derivation of the $X^2$-pdf for k degrees of freedom in the following article: http://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution or give ...
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2k views

95% confidence interval around sum of random variables

Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not ...
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82 views

How to obtain a pdf of a random variable defined as a function of many variables?

Given $N$ independent random variables ($X_1$,$X_2$,...,$X_N$) with individual pdfs $f_1$,...,$f_N$: How to determine the pdf of a random variable $Y=G(X_1,...,X_N)$?
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119 views

econometric transformation of variables

I would like to ask a question, which has been already asked, however I dind´t fully understand the answer and nobody replied to my edit question..maybe you can give me a arithmetic example for one ...
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1answer
190 views

How is the Standard Error and the Unbiased Estimate of the Variance Related?

I am going through a statistics textbook and there are two similar formula's that I cannot seem to grasp, one under the "Sampling Error" section and the other under the "Unbiased Estimator" section. ...
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47 views

econometric data

I have a quit straight forward question: I have a variable which is coal rents measured in 2009 US Dollar. I would like to set this variable in relation (ratio) to another variable which is PPP ...
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112 views

What is the probability of selecting an item of a list of items that have not been selected before?

Story behind my question From time to time, I like to play DotA. It is a competitive computer game, where you pick 1 out of more than 100 heroes and play against each other. For simplicity, let's ...
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47 views

Help with joint distribution functions.

Suppose $f_{XY} (x,y)=f_{1}(x,y)+f_{2}(x,y)$ where $f_{1}(x,y)=\frac{1}{198}$ for $0≤x≤3,0≤y≤2$ and $0$ otherwise. and $f_{2}(x,y)=\frac{1}{66}$ for for $0≤x≤8,0≤y≤8$ and $0$ otherwise First part ...
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88 views

Am I Calculating Quartiles Correctly?

Here is my data set: 5, 8, 12, 42, 43 I am trying to work out the first quartile. When using R (the programming language) it shows the 1st quartile as 8, but when I work it out manually, it shows it ...
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41 views

Confidence intervals for the variance. What if data is not noramlly distributed?

I am writing an essay about confidence intervals for the variance and there is a lot of information available under assumption that our data is normally distributed, but there is not much said about ...
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253 views

Show gamma-function $\Gamma(r)$ is well-defined for any $r >0$ (the limit of the improper definite integral exists)

Introduction: I have proved the following: Suppose that Poisson events are occuring at the constant rate of $\lambda$ per unit time. Let the random variable $Y$ denote the waiting time for the rth ...
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144 views

Asymptotic efficiency of maximum likelihood estimate

Let us consider a simple statistical model $\{f_{\theta}\}$ where $\theta\in U$, an open subset of $\mathbb{R}$. Let $X_1,\dots,X_n$ be sample drawn from $f_{\theta}$. I know, under some regularity ...
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471 views

Need help with expectation of summation for power of Gaussian variable.

I am trying to derive a formula, and getting stuck on a part of the derivation. Basically what I have is the following: $$ P = \sum_{n=0}^{N-1} \sum_{k=0}^{N-1} \mathbb{E} \Big[x^2[n] x^2[k] \Big] $$ ...
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33 views

Prediction model from input/output observations

I want to make a prediction model for the following transformation $f:\mathbb{N}^N \rightarrow \mathbb{R}^M$ $N$ is around 20 and vectors $\mathbf{v} \in \mathbb{N}^N$ are sparse with $\sum_{i=1}^N ...
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69 views

what is the bias of an estimator

The point estimator $\hat\theta$ of a parameter $\theta$ is some function of the sample $D=\{x_1,...,x_n\}$, $$\hat\theta=g(D)$$, since $\hat\theta$ depends on the sample $D$ we're using, so $\hat\...
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2answers
82 views

Fitting curve to more complicated exponential

I've got a conjecture about the relationship of a dataset. My intuition says it's a decaying exponential, but I want to know which a and b in $e^{-ax^{b}}$ best fits the data. What's the best way to ...
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3answers
134 views

Esoteric knowledge regarding statistic tests like $F$-test, $t$-test and $X^2$ (Chi-Square) etc.

For a year or two I've been doing/learning statistics using books written both for engineers and semi-professionals. I know how to apply most of the theory that statisticians use on job, but I'm still ...
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1answer
60 views

(Second) derivation of J.R. Gott's probability distribution

I'm trying to digest J.R. Gott's 1993 Nature paper describing how the Copernican principle can be applied to estimate the duration of our species (and many other neat things). The paper opens with a ...
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1answer
856 views

Mean and median of a linear piecewise pdf

I am studying for a test and I was wondering what is the mean and median of any linear piecewise pdf and why. Also just so you know p.d.f is probability density function.
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195 views

What effect does repetition or momentum have on probability?

Let's say you have thirty right answers in a row, what is the probability of having another right answer? Is there a name for this principal? Can you give a reference?
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50 views

calculate the percentage of workers getting wages between $22$ and $58.$

I want to calculate the percentage of workers getting wages between $22$ and $58.$ I don't know what to use. Please help me. $$\begin{cases}Wages&&No~of~Workers\\0-10&&20\\10-20&&...
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book related query

I have been solving a lot of problems in algebra, calculus, probability and statistics. But have never encountered problems that consist of every mathematical field mentioned above (at max two ...
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635 views

finding $E[X]$ of a exponential distribution with a deductable

Losses follow an exponential distribution with parameter $\theta$. For a deductible of 100, the expected payment per loss is 2000. What is the expected payment per loss for a deductible of 500? They ...
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180 views

Moment generating function for the sum

Let $r_i, i=1, \ldots, n$ be Rademacher random variables, i.e. taking values $\pm1$ with probability $\frac 12$ and such that deference of them is equal to $ D$. Let $x_i, i=1, \ldots, n$ be real ...
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48 views

Determine Probability of Numbers Within a Set, Uniformly Distributed, to Sum to a Number

I have a set of numbers, for example (0,2,3,4,5,6), each with even distribution. I want to know what the chances are that some number of them (X) will add up to Y or greater. What would that formula ...
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3answers
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combinatoric question - balls arrangement

In how many ways can you arrange 200 balls into 40 cells, such that the sum of balls in cells 1-20 is greater (not equal) the sum of balls in cells 21-40? So, the number of posibilities can be ...
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1answer
45 views

Quantile density function: student's t vs normal

I'm trying to show that $\frac{\exp\Big(\frac{\Phi^{-1}(x)^2}{2}\Big)}{\Big(1+\frac{\Psi^{-1}(x)^2}{\delta}\Big)^{(\delta-1)/2}}$ goes to infinity as $x$ goes to one, where $\Phi^{-1}$ denotes the ...
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439 views

Insurance deductable problem

Suppose the random variable $X$ represents the amount of damage done to a rusty Pontiac Sunfire during the 2007-2007 year. Ninety-two percent of the time, the vehicle will sustain no damage. Otherwise ...
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1answer
56 views

Least-Squares Estimator in a Two-Way Layout

I'm having a bit of trouble with the following problem in statistics (particularly ANOVA): Consider a two-way layout in which the effects of the factors are additive; suppose also that there are $K_{...
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72 views

Neyman–Pearson lemma for non monotonic spaces

Question: Does the Neyman–Pearson lemma give instructions for how to construct the test when the outcome space is not monotonic? I suspect the answer is NO, but I would like to: Get an affirmative ...
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158 views

Finding the variance of a statistic.

$X_1,\cdots,X_n$ are independent random variables from $N(\mu,\sigma^2)$ distribution. Define $$T=\frac{1}{2(n-1)}\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$$ I have shown that it is an unbiased estimator of the ...
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Multiple regression degrees of freedom $f$-test.

I'm finding conflicting information from college textbooks on calculating the degrees of freedom for a a global $F$-test on a multiple regression. To be absolutely clear, assume there are 50 ...
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148 views

Finding a PDF given the MGF

If I am given a MGF of any probability density function, say $\dfrac{1}{1-t}$ How can I find the PDF? What would the idea be behind this?
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126 views

The sum of Gaussian functions

Suppose there is a normal distribution and the Gaussian function is $F(x)=\exp(-c\|x-b\|^2)$ where $c$ is a constant and $x,b\in \mathbb{R}^N$, b means the mean value. $F(x)=\exp(-c\|x-b\|^2)=F(x)=\...
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Hypothesis Testing-Statistics

$X_1$ and $X_2$ constitute a random sample of size 2 from a population given by $f(x, T)=T\cdot {x^{T-1}}$ for $0<x<1$ and equal to zero elsewhere. If Critical region is $x_1\cdot x_2\geqslant 3/...
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2answers
58 views

Is this definition of a quantile proper?

I need to find a proper definition of a quantile. It says: a p-th quantile $x_p$ is a number, that satisfies the following conditions: $$ 0<p<1 $$ and $$ P(X \le x_{p}) \ge p $$ and $$ P(X \ge ...
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202 views

Probability Question (Equally likely sample points)

An airport bus deposits 25 passengers at 7 stops. Each passenger is as likely to get off at any stop as at any other, and the passengers act independently of one another. The bus makes a stop only if ...
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79 views

A Question About Sampling

I have a large population of size n from an unknown continuous random variable X, and I do not know the underlying distribution of X. Suppose that I know the minimum sample size b required to ...