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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Why normal approximation to binomial distribution uses np> 5 as a condition

I was reading about normal approximation to binomial distribution and I dunno how it works for cases when you say for example p is equal to 0.3 where p is probability of success. On most websites it ...
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0answers
4 views

Beta marginal distributions and Dirichlet distribution

I know Dirichlet distribution has Beta marginal distributions (sum to 1). I am not sure if the other direction is also correct. That is, for example, if we have 3 beta distributions: $X_1 $ ...
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1answer
15 views

Small population $t$-test

If the population is small, can I use the normal distribution or I need to use $t$-student like if it was a sample? eg: In a class of $8$ students, $4$ male and $4$ female, I need to compare the ...
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2answers
19 views

Distribution of ratio of uniform and exponential random variables

This is a homework question, I feel like I'm doing it right, but I can't seem to get the answer to match up. I have a uniform RV from 2 to 4, and an exponential with mean 4, so $X \sim ...
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2answers
13 views

How to weight three different variables to create a ranking

I have data on the graduation rate, tuition, and average student income (a measure of accessibility) for about 7,000 colleges and universities in the U.S., and I'm trying to compose an interactive in ...
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12 views

Normal distribution exercise [on hold]

In a factory, compacts are filled with a cosmetic powder. We consider the weight of the powder follows a normal distribution $N\sim(\mu, 1.21)$. The value of $\mu$ depends on the setting of the ...
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1answer
55 views

Is there any statistical method to compare two curves?

Is there any statistical method to visually compare two curves? What is the best and correct way to compare two similar curves and calculate the error/difference in percentage? I have created a ...
-3
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0answers
12 views

Help defining the statistical number that pi posses a philosophical question in the first 15 characters. [on hold]

I am looking for assistance calculating the statistical possibilities that pi posses a philosophical question of life in the first 15 characters. Additionally, it has the Fibonacci Sequence and the ...
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17 views

Find spikes in data

I have some datasets and I need to find spikes in them. Imagine the data looks like trading data. If the spike is big enough, I need to log it, otherwise, proceed in the analysis. I tried with a ...
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1answer
14 views

Functional Choice for p in a Bernoulli Distribution

Why is the functional choice $p = \exp(x)/(1+\exp(x))$ to model $p$ a good one in a Bernoulli distribution? Is it because it is limited at $0$ as $x$ approaches $0$ and $1$ as $x$ approaches ...
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1answer
20 views

probability and applied statistics 3 [on hold]

given two urns, suppose urn 1 contains four black and seven white balls. urn 2 contains three black , one white , and four yellow balls . we select an urn at random and then draw a ball . what is the ...
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0answers
12 views

Estimating variance in least squares in regression

How to show $\hat\sigma^2=$$1\over n-2$$[S_{yy}-$$S^2_{xy}\over S_{xx}$]. I don't know the matrix form in regression.I don't even understand how to begin this
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Given two players competing, what is the probability of one getting ahead x wins vs the other getting ahead y

We have 2 players, A and B, competing. The probability that A wins a match is p, making the probability that B wins a match (1-p) = q. The game is won by player A as soon as he gets one more win than ...
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2answers
20 views

Estimate standard deviation of sample

You have a random sample of 25 objects with mean weight of 24 grams, estimate the standard deviation of the sample. In addition, you know it's supposed to be 25 grams with a deviation of 1 gram, but ...
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1answer
20 views

properties of least square estimators in regression

$Y_i=\beta_0+\beta_1 X_i+\epsilon_i$ where $\epsilon_i$ is normally distributed with mean $0$ and variance $\sigma^2$ . The least square estimators of this model are $\hat\beta_0$ and $\hat\beta_1$. ...
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0answers
72 views

Is this method to find mean already discovered?

I am a 10th class student and in our syllabus, we have three methods for finding mean of grouped data: Direct method. Assumed mean method. Step deviation method. Out of these, the Step deviation ...
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15 views

Confidence Intervals Theory

The problem in the textbook is: Let $0 \le γ \le α$. Then a $100(1 – α)\%$ CI for $μ$ when $n$ is large is $$ \left(\bar{x} – zγ\frac{s}{\sqrt{n}}, \bar{x} + zα-γ \frac{s}{\sqrt{n}}\right) $$ ...
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2answers
38 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
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1answer
27 views

If $X$ ~$Uni(-1,1)$ show that $X$ and $X^2$ are not independent

I provide my approach in solving this but I amd not entirely sure whether I am correct. Since X~uni(-1,1) $f_X(x)=1/2$ and $F_X(x)=(x+1)/2$. $F_{X^2}(x)$=$Pr[X^2≤x]$=$2F_X(√x)$=$(√x+1)/2$. Hence ...
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0answers
14 views

confidence coefficient z value

I'm having a bit of difficulty understanding a concept in my notes and was wondering if someone could help me. This is probably a really simple concept that I've just completely overcomplicated but ...
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0answers
10 views

Writing in Lag form and finding the characteristic polynomial, MA(2) with constant

I'm wondering how to write an MA(2) model with a constant in lag form such that I can calculate the characteristic polynomial and get the roots (to see if it's stable). The model is given by $Y_t = ...
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0answers
15 views

Statistic test, t-test,binomial sign test , Pearson and Spearman

I have the following situation: (i) It was observed that the average PSI on rainy days is lower than the average PSI on sunny days (PSI is a measure of air pollution). (ii) After introducing airbags ...
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1answer
25 views

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
23 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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5 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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29 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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0answers
15 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
29 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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0answers
16 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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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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2answers
44 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
18 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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1answer
34 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
34 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
39 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
94 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 ...
0
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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
45 views

Joint PDF and Conditional PDF [on hold]

$X$ 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
16 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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0answers
38 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
55 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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0answers
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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
35 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, ...
0
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0answers
29 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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30 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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24 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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14 views

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 ...