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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formulating lpp

A firm produces three types of products viz., $A$, $B$ and $C$, which are processed on three different machines viz., $M_1$, $M_2$ and $M_3$. The time required to process on unit of each of the ...
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12 views

When would you use a triangular or box kernel instead of gaussian?

Just a conceptual question regarding density estimation. Empirically, the gaussian kernel gives me lower MISE values than triangular or box. Epanechnikov gives me the best MISE values if the ...
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20 views

Establish $\hat{\sigma}^{2}=\frac{1}{n-1} \sum_{i=1}^{n} (Y_{i}-\hat{Y}_{i})^{2}=\frac{1}{n-1}\sum_{i=1}^{n} (Y_{i}-bx_{i})^2$ is unbiased

Establish $\hat{\sigma}^{2}=\frac{1}{n-1} \sum_{i=1}^{n} (Y_{i}-\hat{Y}_{i})^{2}=\frac{1}{n-1}\sum_{i=1}^{n} (Y_{i}-bx_{i})^2$ is an unbiased estimator of $\sigma^{2}$ Note: This is coming from a ...
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Multinomial/Binomial MSE

Let $T_{1}=\sqrt{\frac{N_{1}}{n}}$ and $T_{2}=1-\sqrt{\frac{N_{3}}{n}}$, where $N_{1}\sim\operatorname{Binomial}(n,\theta^{2})$ and $N_{3}\sim\operatorname{Binomial}(n, (1-\theta)^{2})$. Compute the ...
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60 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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74 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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78 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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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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50 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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63 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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53 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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47 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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27 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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267 views

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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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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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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286 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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50 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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30 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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90 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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50 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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36 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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25 views

How do I apply this probability transformation formula?

This is a simple formula but I'm struggling with it. I want to apply $$ f_{Y} (y) = \frac{\left(f_{X}(\sqrt{y})+ f_{X}(-\sqrt{y})\right)}{\sqrt{2y}} $$ to $f_{x}(x)= \frac{1}{3}$ for $-1<x<2$, ...
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27 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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73 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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102 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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33 views

Discrete Time Markov Chain Probability Question

I just wanted clarification for the probability in a DTMC. I know this conditional probability with 3 variables if S = {a,b,c}: $$ P(X_1=a,X_2=b|X_3=c) = P(X_1=a|X_2=b,X_3=c)P(X_2=b|X_3=c) $$ but ...
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2answers
113 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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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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3k 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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287 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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51 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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33 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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118 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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39 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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Probability question! with and without replacements.

Urn contains $10$ blue balls, $5$ red balls, $5$ green balls. When $9$ balls are selected, what is the probability of $7$ balls being blue? i) with replacement ii) without ...
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94 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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108 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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calculate direct and indirect path coefficients

suppose we have following data wth mean and standard derivation corresponding ...
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50 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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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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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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105 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 ...
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Help in mathematical combinatorics.

Hi guys I am studying for my exam which is in a few hours and I ran into two past exam problems. Questions: 1) how many 7 letter sequence you can make with a,b,c such that there is at least one b ...
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163 views

Show that the LS estimator b is unbiased for $\beta$ when regressing without intercept

Okay so I have gotten down to $b=\beta + \frac{\sum_{i=1}^{n}x_i \epsilon_i}{\sum_{i=1}^{n} x_{i}^{2}}$ but I cannot figure out how to show that second term is 0.
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67 views

How to find expected value?

$P(X=1,Y=1) = 1/3 $ $P(X=2,Y=1)=0 $ $P(X=3, Y=1)=1/6 $ $P(X=1,Y=2) = 0 $ $P(X=2,Y=2)=0 $ $P(X=3, Y=2)=0 $ $P(X=1,Y=3) = 1/6 $ $P(X=2,Y=3)=0$ $P(X=3, Y=3)=1/3 $ If I am given the following ...
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Statistics over a flow of data

The story: The IP protocol used for Internet routing breaks each flow to many packets. The router handles a stream of intermixed IP packets from different flows. The statistic collection aims to ...
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42 views

Doubt on an Adaptive algorithm used for Voice Activity Detection

Recently in the task of implementing a adaptive energy algorithm for VOICE ACTIVITY DETECTION ,i came across an algorithm which is working properly and well but the problem is i am not able to ...
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90 views

Calculate the error given a tolerance

I have a noob statistics question. Is there a function, such that given the residuals from the line of best fit, and a probability, A, it will return B such that there is an A probability of being ...
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392 views

Calculating the probability of an event occurring in a specific time period

I am confused at how to approach the following question, i.e. what probability formula I am supposed to use. If the probability of a flood is 0.12 during a year, what is the probability of two floods ...