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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Looking for a statistics formula to show deviations from zero

I'm looking for a formula to show deviations from zero. I have a data set with values that that is normalized to positive real numbers where zero is perfect and higher is worse. I need a statistical ...
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1k views

What is the difference between the Bernoulli and binomial distributions, if any?

I am working problems for statistics, and I am getting somewhat mixed up with Bernoulli and binomal distributions. I know that the binomial distribution is a number of n independent Bernoulli trials. ...
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85 views

Exclude an RGB color from a set

I'm currently implementing an algorithm to split an image into smaller chunks, based on straight line separators. Here's the image I'm processing. It's very small, so you may want to save it and ...
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258 views

MLE of a function involving absolute values

I have some pdf $f(x|\theta) = (\theta/2) ^ {|x|} \cdot (1 - \theta) ^ {(1 - |x|)}$ where $x = -1, 0, 1$, and $0 \le \theta \le 1$. I am tasked with finding the MLE of theta. The way I see it there ...
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184 views

How big of a sample size is necessary to be sufficiently confident in predictions?

A doctor at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be $90\%$ confident that her estimate is within $2$ ...
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56 views

How to derive the marginal probability function of X?

Let $X$ and $Y$ be discrete random variables with joint probability function $f(x,y)=k\frac{2^(x+y)}{x!y!}$ for $x=0,1,2..$ and $y=0,1,2...$,where $k$ is a positive constant. The answer is ...
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70 views

Histograms with frequency density on the vertical axis

In a histogram with frequency density on the vertical axis, could the actual frequencies for each bar of the histogram come out not as whole numbers?
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48 views

Unbiased estimator of standard deviation

Let $X$ be a random variable of distribution $N(\mu , \sigma ^2)$, where we know the value of $\mu$ and we don't know the value of $\sigma$. My task is to choose number $d$, such that random variable ...
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59 views

Calculation the Standard Deviation

I want to calculate the standard deviation of the following numbers: 30, 45, 45, 60, 75, 80, 90, 100, 110, 120. As far as I know, that would be ...
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2k views

Convert nonlinear regression equation to a linear regression equation

The question is: "Show how the nonlinear regression equation y=aX^B can be converted to a linear regression equation solvable by the method of least ...
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787 views

Find E{1/x} if we are given a density function with continuos random variable

Let X be a continuous random variable with density function $$f(x) = \begin{cases}\frac{x}{30}(1+3x) & 1 < x < 3 \\0 & \text{otherwise}\end{cases}$$ Find $E\left(\frac1x\right)$
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294 views

Find a transformation such that the Fisher Information is constant in terms of the parameter.

Say you have a Gamma distribution with parameters $\alpha$ known and $\theta$ unknown. Find a transformation of $\theta$, $\eta=g(\theta)$ such that ${\cal I} (\eta)$, the Fisher Information, is ...
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210 views

What approximately is the probability that she will lose 2 or more pieces of luggage

According to an airline report, roughly 1 piece of luggage out of every 2000 that are checked is lost. Suppose that a frequent-flying businesswoman is checking 1200 bags over the course of the next ...
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373 views

Determine expected winnings if a game is played where a fair coin is tossed until the first tail occurs

A game is played where a fair coin is tossed until the first tail occurs. The probability $x$ tosses will be needed is $f(x) = 0.5^x$, $x = 1,2,3,\dots\;$. You win $2^x$ if $x$ tosses are needed for ...
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1answer
49 views

Mean and Variance of probability distributions

I know how to calculate mean and variance of some given numbers but I have trouble computing them for probability distributions especially when it is a continuous probability distribution. For ...
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137 views

How to find “approximate most common” value from a list of RGB values

I have about 50 equally sized photos of magazine covers, which I'm attempting to blend into one composite image that shows the "average" cover. Each of the covers has a single face on it, so the ...
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1answer
197 views

Numeric approximation for fitting a Gamma Distribution with a single parameter

Given a series of $N$ observations $\left(x_1, \ldots, x_N\right)$ that follow a Gamma distribution with a single parameter, $ \text{Gamma}(k, k)$, what is the maximum likelihood estimate of $ k $?. ...
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144 views

Theoretical impossibility? Deviation from normality with a sample greater than 300?

Huge thanks in advance! I've been lead to believe that the following is a theoretical impossibility: a population larger than 300 records without an approximation of a normal distribution. The ...
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63 views

Calculating Bayes factor

Example: Integer-valued data $y = (y_1, ...,y_n):$ $M_1 = Geometric(\theta_1)$ likelihood with $Beta(\alpha_1, \beta_1)$ prior on $\theta_1;$ $M_2=Poisson(\theta_2)$ likelihood with $Gamma(\alpha_2, ...
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109 views

Best fit line using geometric distance (not vertical distance)

There must be a theory of finding the best fit line to a bunch of points in the plane, where "best fit" is defined by the geometric distance, not vertical distance. In other words, we are trying to ...
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1answer
86 views

Likelihood functions

Suppose that $Y_1,\ldots, Y_n$ are independent and identically distributed random variables with density function $$f(y \mid \theta) = \frac{\theta^2}{y^2} e^{-\theta/y}, \ \text{ where } \ y, \theta ...
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96 views

Probability from a Moment Generating Function

Is there a way to calculate probabilities given a MGF? Or, does there exist a method to obtain a probability density function from a MGF? Here is the problem I am interested in solving: Find the ...
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68 views

Conditions for non-decreasing conditional expectation

Let $X$, $Y$ and $Z$ be three real random variables. I would like to know if assuming Regression Dependence * , in the sense that $\Pr[Z\leq z |Y=y]$ is non-increasing in $y$, is sufficient or ...
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1answer
34 views

Assume $Y_i=\beta x_{i} + \epsilon_{i}$ What is the variance of the LS estimator b?

So far I have $b=\frac{\sum_{i=1}^{n} x_{i}y_{i}}{\sum_{i=1}^{n} x_{i}^{2}}$ So I substituted for $Y_i$ and got $Var(b)=Var(\beta + \frac{\sum_{i=1}^{n} x_{i} \epsilon_{i}}{\sum_{i=1}^{n} ...
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Finding expected payout

A life insurer has created a special one-year term insurance policy for a pair of business people who travel to high risk locations. The insurance policy pays nothing if neither die in the year, ...
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25 views

Statistics PDF problem

Let Xsub1, Xsub2, ..., Xsub48 be a random sample of size 48 from the distribution with pdf f(x) = 1/(x^2), 1 < x < infinity. Approximate the probability that at most 10 of these random variables ...
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151 views

Finding a moment generating function given E(X) and E(x^2)

I am trying to find the moment generating function. It takes values in the set {0,1,2} with moments E(X) = 1 and E($X^{2}$) = $ \frac 3 2 $ I know then that M'(0) = 1 and M"(0) = $\frac 3 2 $ I ...
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1k views

K Nearest Neighbor Density Estimation

An intuitive way to estimate the pdf of a distribution $f$ is described here. Given a set of points you find the distance to the $k$th nearest neighbor for a point $x$ that we want to know the value ...
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65 views

Combination of Conditional Expectations

Let $(T,S,\theta)$ be random variables in $\mathbb{R}^3$ with joint pdf noted by $f_{T, S ,\theta}(\cdot)$ I want to know if $E[\theta|T\geq t,S\geq s]= \frac{\int_{-\infty}^{\infty} ...
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40 views

A question on the formula in calculating the standard score

We all know that the formula for calculating z, the standard score, from the raw data x is given by z(i) = [x(i) – μ]/σ {Some use ‘bar x’ instead of μ but μ is much easier to type and this is not ...
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125 views

Finding the MLE of a function when L'($\theta$) doesn't depend on $\theta$

Here's the problem: Find the MLE of of $\theta$ when $f(x\mid\theta)=(1+x\theta)/2$ for $-1<x<1$, $=0$ otherwise. $0<\theta<1$ Find the maximum likelihood of $\theta$ and find its ...
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147 views

Finding probability of uniform random variable given a condition with another random variable

Suppose X and Y are independent and uniformly distributed on the unit interval (0,1). Find: $$P[Y>\frac{1}{2}\,|\,Y>1-2X]$$ How I approached it was to find the area where $Y > 1 - 2X$, and ...
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42 views

Number of possiblities - N choose K?

I have $N$ particles, with $m$ particles in state 1 and $N-m$ particles in state 2. These $N$ particles are placed in $N_s$ sites, where $N_s > N$. What are the total number of states or ways this ...
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1answer
32 views

Non-close-form Regression Research

As I try to process some physic experiment data that I don't have the closed form formula with unknown parameters, I have to use some regression models like polynomials or normal distributions . The ...
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140 views

Discrete Uniform random variable calculate mean and var

A lottery player decides to use a random variable generator to help him decide how many tickets to buy. He generates a discrete uniform random variable N taking values 1 through 4 with equal ...
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1answer
160 views

Mean and Var of a gamma distribution

Let X have a Gamma distribution with a known scale parameter 1, but an unknown shape parameter, that itself is random, and has the standard exponential distribution. How do I compute the mean and ...
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2answers
238 views

Proof that distribution has power law tails from having infinite moments

Is the fact that the 2nd (or higher) moment of a distribution is infinite (while, say,the first moment is finite) proof that the distribution has power law tails? Thank you in advance.
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102 views

Formulating regression model in matrix notation

The observations $y_1, y_2, y_3$ were taken on the random variables $Y_1, Y_2, Y_3$ where $Y_1=\theta+e_1$ $Y_2=2\theta - \phi+e_2$ $Y_3=\theta +2 \phi+e_3$ and $E(e_i)=0, var(e_i)=\sigma^2 ...
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1answer
13k views

How to calculate the covariance matrix

I tried searching a lot on the net and got the following sources: Source One Source Two The first source seems to be incorrect cause when I calculate it using matlab it comes to be different from ...
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1answer
2k views

MLE of fourth moment of normal distribution

Take $X\sim N(0,\theta)$, and let $\phi = E(X^4)$, the fourth moment. What is its MLE, $\hat{\phi}$, and what is the asymptotic distribution of $\sqrt{n}(\hat{\phi} - \phi) $ as $n\to \infty$? Any ...
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1answer
68 views

The Average Speed of an object

I'm pretty sure this has more to do with fundamental Math than Physics and that is why I'm asking this here rather than Physics.SE Imagine some object travelling along a straight path from point $A$ ...
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1answer
36 views

Difference of a likelihood function for a vector and a single value

$p(x\mid C)$ is defined as the probability density of a point $x$ given that it belongs to a class $C.$ But what of $p(\mathbf{x}\mid C)$ where $\mathbf{x}$ is a vector? I'm finding hard to ...
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1answer
220 views

Regarding Transformation on Uniformly Distributed Unit Disk

$(X,Y)$ is distributed uniformly on the unit disk. The transformations are: $$ Z = {X + Y \over \sqrt{2}}\,,\qquad W = {X - Y \over \sqrt{2}} $$ I solved these equations in terms of $X$ and $Y$ ...
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43 views

How to start statistical analysis paper

I am literally planked ,i dont know what to do. The question is how obesity is linked to hospitalization of an individual. so we have the data set of subjects according to their gender, if they had ...
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130 views

Finding the expectation of functions of random variables with a bivariate normal distribution

X and Y have a bivariate normal distribution. I am given that $E[X] = 4$ and $E[Y] = 10$. I am asked to find $E[X^2 - Y^2]$ WITHOUT integration. I know how to solve for this using integration, but ...
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180 views

Given the joint distribution of two random variables, compute the probability that one is less than the other?

Let $X$, $Y$ have the joint density function $$f(x,y) = \frac{1}{2\pi} e^{-(x^2+y^2)/2}$$ Compute $P(X<Y)$. I believe that I should set up a double integral over this function, like so: ...
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1answer
59 views

Independent sequences

Let $\{x_i\}_{i = 1, ...,n}$ , $\{y_i\}_{i = 1, ...,n}$ be sequences generated by a pseudo-random number generator using different seed keys, for example $ x_0$ and $y_0$. Are $\{x_i\}$ and $\{y_i\}$ ...
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28 views

How can i simplify the following term to get the right side?

$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}=\frac{1}{n}\sum_{h=1}^{L}{(W_hS_h)}^2$$ where, $n_h=\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h$ $\quad\text{and}\quad$ $W_h=\frac{N_h}{N}$ $\quad\text{and}\quad$ ...
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175 views

Strong Law of Large numbers, prove expression is Standard Normal

Question: "Let $X_{1},X_{2},\cdots$ be a sequence of independent random variables such that $X_{n}$ is binomial with parameters $2n-1$ and $p=\frac{1}{2}$. If $$Y_{n} = \frac{2(X_{1}+X_{2}+\cdots ...
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315 views

least squares regression parabola

In my AP Stats course, we just finished our chapter on least squares regression lines and are moving on to non-linear regressions. I was expecting a least squares regression parabola, but instead we ...