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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156 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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226 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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98 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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11k 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
1k 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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67 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
35 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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202 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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92 views

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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127 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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168 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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1answer
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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1answer
171 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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1answer
291 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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193 views

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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318 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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79 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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1answer
213 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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3answers
3k 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 ...
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1answer
87 views

Finding the pdf of product of uniform random variables [closed]

Finding the pdf of product of $U_i$ from $i=1$to $n$ where $U_i$ are independent uniform$(0,1)$ random variables.
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175 views

What is the moment generating function and variance

Find the moments of the random variable $X$ if its moment generating function is $$M_X(t) = (1−p_1 −p_2)+p_1e^t +p_2e^{2t}.$$ What is the variance of $X$?
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76 views

Colored blocks and towers

George got a big box from his parents. In this box are colored blocks. He has white, black, red, blue and orange blocks. These blocks are all exactly the same size and of he has the same amount ...
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1answer
59 views

Random variables $X$ and $Y$ that are not independent

I am trying to solve a stat question, here is the question: Give an example of two random variables $X$ and $Y$, each taking values in the set $\{1,2,3\}$ such that $P(X = 1; Y = 1) = P(X = 1)P(Y = ...
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144 views

Hypothesis Testing, simple against composite.

i've got some trouble understanding this exercise from Amemiya's Introduction to statistics: Given the density $f(x) = \frac{1}{\theta}$ from $0 < x < \theta$, and 0 elsewhere, we are to test ...
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10k views

The expectation of absolute value of random variables

I need some help with the following problem: Let $X_1,...,X_n$ be a random sample from Normal$(0,1)$ population. Define $$Y_1=| {{1 \over n}\sum_{i=1}^{n}X_i}|, \ Y_2={1 \over ...
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31 views

1-F(x) as F(x) goes to 1

I stumbled into the following statement and I am not sure how to prove it. The statement is : Given a distribution function $F(x)$ we can represent the survival function $1-F(x)$ as $-logF(x)$ for ...
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143 views

Expectation and variance of occurrences of patterns in a random text

I'm not sure how to complete the following problem from the book An Introduction to Bioinformatics Algorithms: Problem 9.1 Derive the expectation and the variance of the number of occurrences ...
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1answer
40 views

Convert list of test results into a comparative score

My students have completed a touch typing test that have given them a WPM (words per minute) result. I now want to give a score to those results out of an arbitrary number, say 10. I will later use ...
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192 views

Unbiased estimators of theta

Suppose $\hat\theta_1$ and theta $\hat\theta_2$ are both uncorrelated and unbiased estimators of $\theta$, and that $\text{var}\hat\theta_1=2\cdot \text{var}(\hat\theta_2)$. a) Show that for any ...
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2answers
71 views

What is the fastest method for factoring this

I want the square root of $.09e^{-2t} + .24e^{-t} +.34 +.24e^t + .09e^{2t}$ I need this to find the moment generating function given it's product with itself (the product of two identical moment ...
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1answer
167 views

What does this notation mean? $f(x|\theta)=\frac{3x^2}{\theta^3}I_{(0,\theta)(x)}$

Particularly I want to know what the meaning of $I_{(0,\theta)(x)}$ is here.
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0answers
26 views

How is a Brownian motion a collection of random variables

In financial mathematics textbooks I have seen it described as a collection of random variables (in the statistics sense of the term random variable), but is that just because at any given moment, ...
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1answer
235 views

Gaussian with a linear combination random variable mean

A very simple (looks like...) statistical problem, however I don't even know how to name it in a formal way... Suppose in a Bayesian framework I have random variables $y, x_1,$ and $x_2$, $$f(x) = ...
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107 views

An inequality involving expectation

Let $f,g$ be two pdfs, and suppose $X$ is a random variable that has pdf $f$. Is it necessarily true that $E[f(X)] \ge E[g(X)]$? Although I doubt this will help, but I got this problem from studying ...
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45 views

Expected length of a random human being is the same as average of the complete human population

Let $n$ be the number of people in the world. And let $X_i$ be the corresponding length in centimeter. I'm trying to prove that $1/n \sum_{i=1}^n X_i=EX_1$. \begin{align} EX_1 &= \sum_{x=0}^{251} ...
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1answer
116 views

Which confidence interval is correct?

I have time series predictions with 50% and 90% confidence intervals. 50% of observations lie below the 50% confidence interval (IIUC, this should happen only 25% of the time), but all observations ...
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1answer
27 views

Calculation of ${\rm E}[e^X]$ where $X$ follows a uniform distribution over $(1,2)$

To compute ${\rm E}[e^X]$, in which $X$ has uniform $U(1,2)$ distribution. Here $f(x)=1$ for $x\in (1,2)$. The formula is $\int_0^1 e^x \,\mathrm dx$. Is the answer $e^2 - e^1$?
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57 views

Complement of A or B

I have a small general question.. Let's say we have two events $A$ and $B$. Is the probability that $A$ or $B$ will happen, the complement of the event that the complement of $A$ and the complement ...
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1answer
161 views

Understanding Poisson Distribution Question?

I have the following question: I have the following formula: However, it is unclear to me how to extract what I need. Would U be equal to 17 for an average of 17 per minute and X be equal to 25? ...
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95 views

If the prior distribution of $\lambda$ is a gamma distribution with mean 1.4 and std dev of .5 find the appropriate values and $\alpha$ and $\beta$

If the prior distribution of $\lambda$ is a gamma distribution with mean of 1.4 and standard deviation of .5 with the form $\pi(\lambda | \alpha, \beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)} ...
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1answer
547 views

Find Bayes estimator of $\theta$

I've got this exercise, which I'm trying to work off using an example, but the example seems very different so I'm not sure if what I'm really doing. I've got a loss distribution for $\theta$: ...
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1answer
201 views

expectation of norm of orthogonal projector

The question has to do with calculating the expected squared norm of a random projection. We have a 2D subspace $T := span\{U1, U2\}$ where $U1$ is a random vector uniformly distributed over unit ...
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1answer
115 views

Find marginal distribution for Pareto prior

I have the following problem: The prior distribution for $\theta$ is distributed $\pi(\theta) = \frac{aP^a}{\theta^{a+1}}$, $\theta >P$ The likelihood for X is uniformly distributed, i.e. ...
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108 views

Can posterior distribution for a continuous variable be greater than one?

This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the following: $$p(\theta|X)=\frac{p(\theta)p(X|\theta)}{\int{p(\theta)p(X|\theta)d\theta}}$$ I ...
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199 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
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224 views

What is the probability that a customer waits for lesser than 3 minutes?

The rate of service is exponential and the service rate is 12 customers served per hour. The arrival of customers is in a Poisson distribution at the rate of 30 per hour. There are 3 servers and the ...
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1answer
44 views

Predict amount of goods to order?

Someone familiar with prediction problems, where you need to predict how much goods to order given some history? For example, this week you have the history of how much apples you ordered, how much ...
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1answer
550 views

Random variable and Poisson distrubtion

Given that $X$ is a random variable having a Poisson distribution, compute the following: (a)When $μ=0.5$, $P(X>3)$, My attempt: $1-e^{-.5}-e^{-.5}-e^{-.5}\cdot0.5-\frac{e^{-.5}\cdot0.5^3}{3}$ ...