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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Taking the standard deviation over multiple assignments?

Lets say that a group of students all take 5 tests, with the average and standard deviations as follows: test 1: mean 43/50, SD 8 test 2: mean 23/30, SD 4 test 3: mean 56/70, SD 12 test 4: mean ...
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1answer
16 views

Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
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3 views

Is it possible to calculate the autocovariance of a DT WSS signal knowing only it's mean and it's linear estimator?

Let's also assume that we know that $C_{xx} [0] = K$ Where $K$ is some constant. I'm trying to figure out if that's possible and if yes how I would get around doing it.
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27 views

Find E[MSLOF]. Please help.

Find the expected mean squares error of lack of fit. Trial: $$SSLOF=\sum_{1}^mn_i(\bar y_i-\hat y_i)^2\\=\sum_{1}^mn_i(\bar y_i-\bar y)^2-\sum_{1}^mn_i \hat\beta_i^2(x_i-\bar x)^2$$ and ...
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12 views

Heavy tailed sum of iid light tailed random variables

I know that to get one, the number of summands has to be random with a heavy-tailed count variable. I am wondering how you prove the resulting sum is heavy-tailed and in particular wondering if there ...
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1answer
21 views

Clique factorization

I'm reading about Clique factorization in wikipedia: http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization but I'm unable to understand this: What is $X_C$ here? Ok I understood ...
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15 views

ancillary statistics and Basu theorem [on hold]

Let X1,X2,.....Xn be a random sample from f(x)=exp(-(x-M)) x > M, use Basu theorem to show that X(1) " the first order statistics" and S^2 are independent ? I have verified that X(1) is a complete ...
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9 views

Can you find a method of moments of Gaussian AR(1)?

This is an exercise from Mathematical Statistics: Basic ideas and Selected topics, Bickel&Doksum, page 141. Gaussian AR(1) model; $X_i = \mu + e_i, i=1, \cdots,n$ $e_i = \beta e_{i-1} ...
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42 views

Probability, chose two skittles, out of 2 skittles left from a bag of skittles with 5 colors.

so me and my friend are studying statistics but we are just stuck on this stupid skittle question we made up ourselves when we tried to guess the colors of the two last skittles so we can see who will ...
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1answer
15 views

Probability and Standard Deviation

Hey I'm confused about how to do this kind of problem. I can't figure out how to find the standard deviation. There are on average 4 tetanus cases reported in the US each month. What is the ...
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1answer
26 views

Covariance of two values

A fair die is rolled twice (independently). Let X1 and X2 be the numbers resulting from the first and second rolls, respectively. Define Y=X1+X2 and Z=4⋅X1−X2. Find the covariance between Y and Z. ...
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33 views

Help in Stats, Joint p.d.f

Let $X$ and $Y$ be random variables that have a joint p.d.f., which is given by the formula $\displaystyle p_{X,Y}(x,y)=\frac{5e^{−5x}}{x}$ when $0< y < x < \infty$, and $p_{X,Y}(x,y)=0$ for ...
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0answers
9 views

Factorizing about an undirected graph [on hold]

When do we say that a distribution factorizes about an undirected graph $G$ with maximal cliques $C$?
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1answer
17 views

Hammersley–Clifford theorem

I'm reading this paper http://image.diku.dk/igel/paper/AItRBM-proof.pdf and I got stuck in page 4 with equation (1) that's based on Hammersley–Clifford theorem. I'm not good in reading set theory ...
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1answer
31 views

Joint p.d.f stats help [on hold]

$X$ and $Y$ are random variables that have a joint p.d.f., given by $p(x,y)=cx^9y^6$ when $0\le x,y\le 1$ and $p(x,y)=0$ for all other $x,y$. Here $c\ge0$ is a constant, which you should find. What ...
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1answer
85 views

jointly normally distributed random variables [on hold]

Suppose that X and Y are jointly normally distributed random variables, each of which is standard normal, and the correlation coefficient between X and Y is equal to 0.4. Find the probability that ...
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1answer
25 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
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1answer
39 views

Joint PDF and Conditional PDF

$X$ and $Y$ are random variables that have a joint pdf, given by $p_{X,Y}(x,y)=4xe^{-x(y+4)}$ when $x,y>0$ and $\ p_{X,Y}(x,y)=0$ for all other $x,y$. Find a formula for the conditional pdf $\ ...
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1answer
15 views

Example of dependent but conditional independent

There are a lot of events that are independent and conditional independent. Is there any events that are dependent but conditionally independent?
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37 views

The joint pdf of random variables $X$ and $Y$

$X$ and $Y$ are random variables that have a joint pdf, given by $$ p(x,y)= \left\{ \begin{array}{l l} c\ x^9y^6 & \quad ;\ 0\leq x,y\leq 1,\\ \\ 0 & \quad ;\text{ for all other}\ ...
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1answer
54 views

Joint pdf random variables

$X$ and $Y$ are random variables that have a joint p.d.f. given by $p(x,y)=2⋅\frac{(x+2y)}{3}$ when $0≤x,y≤1$ and $p(x,y)=0$ for all other $x,y$. Find the probability that $X<(1/3)+Y$. I'm ...
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1answer
29 views

Probability of Multiple Random Variables

Let $X_1$,$X_2$,$X_3$,$X_4$ be independent standard normal random variables and $Y=X_1^2+X_2^2+X_3^2+X_4^2$. Find the probability that $Y\leq 3$. I thought that you would be using some kind of ...
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Why is $1 - \operatorname{P}(\text{Type I error}) =\overline{\operatorname{P}(\text{Type II error})}$?

I understand that in hypothesis testing, an increase in $\operatorname{P}(\text{Type I error})$ will lead to a decrease in $\operatorname{P}(\text{Type II error})$. However, why does an addition of ...
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1answer
34 views

How to come up with a probability distribution knowing the mean value? [on hold]

I would like to know about some algorithms or techniques to find a discrete probability distribution knowing the mean value. Let's say given the mean=2.5. The probability distribution can be $x_1=2, ...
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28 views

$E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$

If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$ E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...
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29 views

Show that $Y_i$ is independent of $Y_j$ for any $i$ not equal to $j$

Let $\{X_1,X_2,\ldots\}$ be independent, identically distributed, absolutely continuous random variables. Let $Y_n=I\{X_n>\max(1< i < n)\}$ for $n=2,3,\ldots$ a) Show that $Y_i$ is ...
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22 views

POISSON DISTRIBUTION between two values help [on hold]

LET x= NUMBER OF FLAWS OF THE SURFACE OF A RANDOMLY SELECTED BOILER OF CERTAIN TYPE. WITH MEAN = USE POISSON DISTRIBUTION to find p(5 please help me with this
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Be accurate within 0.2 of a percent using gamma distribution

Carrying out enough simulations of an experiment in gamma distribution, how can I find an answer to be accurate within 0.2 of a percent? (my answer is in the form of a percentage) I run an experiment ...
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11 views

Bayesian mean square error

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
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1answer
22 views

$X$ and $Y$ have a joint distribution density function. Working out a marginal density function for $X$ and $Y$

$f_{X,Y}(x,y) = \frac{3}{2}(x^2+y^2)$ if $0 \lt x \lt 1$ and $0 \lt y \lt 1,$ or $0$ otherwise. I want to find the marginal probability density function of $X$ and $Y$ and then find $Pr(0 \lt x \lt ...
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1answer
37 views

Probability of the sum of independent standard normal random variables

Let $X_1, X_2, X_3, X_4$ be independent standard normal random variables and $$Y = X_1^2 + X_2^2 + X_3^2 + X_4^2$$ Find the probability that $Y \leq 3$. For this problem I know that the ...
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1answer
12 views

Expected value, variance and probability from a joint distribution function

Lets say I am given the following table that shows the joint probability function of X and Y: $$\begin{array} \\{}&y=1&y=2&y=3 \\x_=1&0.1&0.2&0.1 ...
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16 views

Expected value of joint p.d.f. with unknown constant

X and Y are random variables that have a joint p.d.f. $$p(x,y)=c*(x^9)*(y^6)$$ when $0<=x, y<=1$ and $c>=0$ is a constant that should be found. What is the expected value of $Y$? I am having ...
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1answer
44 views

Joint density of two functions of random variable

This is online homework, and I'm not always clear on which chapter questions are from, so I might be completely off base. I have two random variables, $X_1$~UNI(5,10) and $X_2$~UNI(4,10), and then ...
2
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1answer
11 views

Joint distribution probabilities

I have a question that is similar to the following(made up here): The construction of a tower of cards is done is two stages, procrastination and the actual building. The time in minutes needed to ...
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1answer
35 views

Roll Dice- Expected Winnings [on hold]

I have a problem like this: At a charity game you pay \$1 to roll a die. If you roll a 6, you get \$5. Otherwise, you get nothing. How do I set up a probability distribution and what is the ...
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0answers
10 views

Contitional Expectation of Sum of two uniform rv

(apologies for posting a more specific version of an earlier question). Let $X$ be uniformely distributed on $[0,1]$ and $Y$ be uniformely distributed on $[m-k,m+k]$ with m being a number in $[0,1]$ ...
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10 views

Statistics P-Value, Critical Value, Null/Alternative Hypothesis

I have a Statistics word problem, that I solved (for the most part, I believe), However I am bad at terminology and figuring out what values go to which question. PROBLEM: When 50 people used the 5-2 ...
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10 views

Dynamic Statistics Equation

There are two data sets, Alpha and Bravo. Every data set in this problem has the same structure. A data set consists of six places (a, b, c, d, e, f) and each place in each data set contains an ...
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2answers
20 views

how to find standard deviation when given a percentage?

I'm stuck on this question, can anyone help? An electronics company's biggest seller is a talking toy. Of the toys produced, 4% have a defective voicebox. A quality control tech randomly collects 250 ...
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10 views

How to calculate the confidence interval of variance [on hold]

n=80 mean=825 SD= 48.5 whats the formula for calculating 95% confidence interval of variance?
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12 views

showing minimum and relative efficiency of 2 independent samples

Let Ȳ1 and Ȳ2 be the means of independent random samples of size n1 and n2 from a normal population with mean µ and variance σ2. a) Show that the variance of the unbiased estimator xȲ1 + (1-x)Ȳ2 ...
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1answer
14 views

unbiased and consistent estimators of 2 independent populations

Let Ȳ1 be the mean of a random sample of size n from a normal population with mean µ and variance σ12 and let Ȳ2 be the mean of a random sample of size n from a normal population with mean µ and ...
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1answer
11 views

marginal distributions of a joint pdf

find the marginal distributions of $$ f(x,y) = 2xe^{-y}, \quad 0 < x,\quad x^2 < y $$ I have drawn the support, with $y = x^2$. Not sure how to proceed (tried it on wolfram advanced calculus ...
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13 views

finding maximum likelihood estimators for gamma distribution

Let Y1, Y2,...,Yn be a random sample of size n from a gamma population with α=2. a) Find the maximum likelihood estimator for β. b) Show that the maximum likelihood estimator for β is an unbiased ...
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26 views

finding maximum likelihood estimators

Let $Y_1, Y_2,\cdots,Y_n$ be independent, identically distributed (iid), each with the following probability density function: $\displaystyle ...
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19 views

finding method of moments estimator for beta distribution

Let Y1, Y2,...,Yn be a random sample from a population that follows a beta distribution with β=3. Find the method of moments estimator of α. Please show all steps and work So far I have equations ...
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1answer
19 views

finding the cdf of the max of a joint distribuion [on hold]

Random variables X and Y have a distribution given by the following pdf: fx,y (x,y) = 1, 0 < x < 1 , 0 < y < 2x ; and o otherwise find the cdf of Z = max(x,y)
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1answer
13 views

Find the probability that the difference between the sample mean and the true population mean wll not exceed 0.5 inch

an anthropologist wishes to estimate the average height of men for a certain race of people. if the population standard deviation is assumed to be 2.5 inches and if she randomly samples 100 mean, find ...
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1answer
25 views

Decomposition of sum of two independent random variables

Let $X$ and $Y$ be two indpenent r.v. How can I get an expression for: $$E[X|X+Y=a]$$ where $a$ is a constant? In other words, is there a general rule to recover the expected value of $X$ when all I ...