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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Regular Conditional Bayesian Experiment
In "Elements of Bayesian Statistics" (1990), Florens, Mouchart and Rolin describe two basic forms of reduction of a Bayesian experiment: Marginalization and Conditioning (Ch. 1). I don't understand ...
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1answer
14 views
Showing uniform convergence in probability
Suppose you want to show $sup_{x\in D}|f_n(x)|\to_p 0$, for $n\to \infty$, where $D\subset \mathbb R$ is a compact interval, $f$ is continuous depending on one or more random variables, and $\to_p$ ...
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1answer
12 views
question on five number summary & quantile.
i know that in five number summary :
25% of a data set lies between Min & 1st quartile.
50% of a data set lies between Min & 2nd quartile, that is, Median.
75% of a data set lies between ...
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1answer
15 views
Calculating the probabilities of different lengths of repetitions of X length numbers
I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
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1answer
15 views
Expectation of function of stochast
I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
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0answers
35 views
Estimating the radius of a circle
I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
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21 views
Neyman-Pearson lemma on Normal distribution
We've got a random sample of iid $X_1,\dots,X_n$. We're testing the mean of $X \sim \mathcal{N}(\mu,\sigma^2)$, where $\sigma^2$ is known. The size of the test $\alpha=0.05$.
$H_0: \mu=0$
$H_1: ...
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1answer
14 views
Means and standard deviations.
Given three independent random variables $X_k$, the means and standard deviations respectively are $X_1 = (A, X)$, $X_2 = (B, Y)$, $X_3 = (C,Z)$.
(1)What is the mean of their sum?
(2)What is the ...
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1answer
17 views
Joint distribution of multiple binomial distributions
In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them.
The original file can be ...
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20 views
Fisher information of a Binomial distribution
The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
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41 views
Probability distribution for a digit of a number
If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
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2answers
28 views
Linear Regression: Expectation Proof
I found the following proof in my notes:
$E(Y_i) = E[\beta_0 + \beta X_i + \varepsilon_i] =\cdots= \beta_0 + \beta X_i$. This does not seem right to me, however. Why would $E(\beta_1 X_i) = \beta_1 ...
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6 views
Model Fit in Logistic Regression
I've fitted a binary logistic regression model with 3 fixed effects (and with 2 random effects, but I don't think that influences my question), and now I want to check how well the model fits the ...
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0answers
20 views
please show that $\hat\mu_i\sim N(\mu_i,\frac {\sigma^2}{n_i})$
Statistical model for Complete Randomized design
$y_{ij} = \mu + \tau_i + \epsilon_{ij}$
where, $i$ denotes treatment and $j$ denotes observation.
$i=1,2,...,k\quad and \quad j=1,2,..., n_i$
...
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2answers
41 views
Integration by parts disconnect
I'm trying to integrate $\displaystyle E(Y^2) = \int^\infty_0 y^2\lambda e^{-\lambda y} dy$
doing it by parts this is my logic.
$\displaystyle E(Y^2) = \int^\infty_0 y^2\lambda e^{-\lambda y} dy$ ...
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1answer
19 views
Finding the MLE of a multinomial distribution (uneven probabilities)
I am trying to simulate loaded die where the face probabilities are:
$$
p_1=p_2=p_3=p_4=1/6+\theta\text{ and }p_5=p_6=1/6-2\theta
$$
And so using the multinomial distribution I have:
$$
...
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3answers
40 views
how to tell whether x and y are independent or not
Suppose that $f_{x,y}(x,y) = \lambda^2 e^{\displaystyle-\lambda(x+y)}, 0\leq x , 0\leq y.$ Find $\operatorname{Var(X+Y)}$.
I'm having trouble with this problem the way to find ...
3
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3answers
37 views
Integration of function help
I'm having problems integrating this function $\displaystyle E(X)=\int^ \infty_0 x\lambda e^{-\lambda x} dx$. I did the integration by parts and had $-xe^{-\lambda x}- \lambda e^{-\lambda x}$. However ...
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0answers
17 views
Marginal Pdfs for Continuous Random Variables
http://oi42.tinypic.com/ddyjph.jpg
this problem is confusing me, i know how to start it, we need to find $f_Y(y)$ so we integrate with respect to x and i get $-2e^{-x}e^{-y}|^y_0$ which then should ...
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2answers
23 views
Is a Relationship Quadratic?
I have a relationship $y=f(x)$ for which I can obtain data through simulation.
I have good reason to suspect that this relationship is quadratic (rather than, say, exponential), and would like to ...
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17 views
Characteristic equation/expression for addtion of n lognormal distributions
$\newcommand{\lognorm}{\operatorname{lognorm}}$
I have to find the expression for (both mean(E[x]) value and error factor 
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1answer
17 views
please prove the following proof related to F distribution.
Suppose $S_1^2$ and $S_2^2$ are two independent unbiased estimate of the common population variance $\sigma^2$ from two random sample of sizes $n_1$ and $n_2$ respectively.
Then show that
...
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1answer
22 views
How do I work out what percent of my customers will be girls and what percent will be boys?
I know that 33.3333% of all girls questions would buy my product and that 80% of all boys questioned would buy it.
What i don't know is how to work out is statistically what percentage of our ...
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14 views
Bayesian Parameter Estimation Doubt
I was going through a pattern recognition book and in the chapter of Bayesian Parameter Estimation I came across this formula. I cannot understand how the 2nd line is derived from the first line. ...
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1answer
27 views
How to compute conditional expectation of a log function
I've been studying the Expectation Maximization algorithm. According to the formula shown here, what I have to do in the M step is to compute a new $\theta$ that maximizes the conditional expectation ...
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0answers
29 views
Hypothesis testing from negative binomial data
For each trial I flip a coin until $3$ heads or $3$ tails occur, whichever comes first. Out of $10$ trials, $3$ trials result in $3$ flips, $3$ trials result in 4 flips, and $4$ trials result in $5$ ...
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1answer
22 views
How does the increase in overall number of events affect the peak (events/time)?
I have a (simple?) question that I hope someone will find interesting enough to help me out with.
A web site has a given number of subscribers who generate a certain amount of traffic on the web ...
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3answers
81 views
A basic question on the definition of $E[X]$
My question is regarding the definition of $E[X]$ in a probability book. It starts with the definition in case of a simple random variable (a random variable which takes only finite number of values) ...
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1answer
24 views
A good reference to learn the concept of partition function
I am looking for a good reference and easy to learn the concept of partition function in mathematics. Can anyone help me?
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42 views
Probability that we choose a two headed coin
We have a $501$ coins on the table, and assume that they have all been flipped onto that table (i.e., there is a mix of heads and tails). This also includes a two-headed coing.
Now if we pick up $1$ ...
3
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1answer
31 views
Please verify this Proportion Hypothesis Test (Statistics)
Questions:
$120$ voters, $39$ identify with radical movement.
does this provide sufficient evidence that more than $25\%$ of voters identify with radical movement?
use $p$-value approach.
My ...
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1answer
39 views
Blackjack card counting, with one whole deck should the “count” end on zero? [closed]
When playing blackjack if you are card counting for a single deck, should the count always come to zero at the end of the deck? Wouldn't it depend on strategy and a corresponding betting table?
Would ...
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1answer
25 views
Combination calculation with reducing set size
My statistics aren't too great, so I'm struggle to work out the result of the following situation.
Say you have 5 sets of 5 possible options (25 options total); and you select 1 option from each set. ...
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1answer
35 views
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1answer
38 views
Cramer-Rao Lower Bound
Assume that $X_1,X_2,\ldots,X_N\sim N(\mu,2^2)$ and $Y_1,Y_2,\ldots,Y_M\sim N(0,\sigma^2)$.
a)Find the Cramer-Rao Lower Bound (CRLB) for the variance of the unbiased estimators of $\mu$.
b)Find the ...
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65 views
Determining a confidence interval for $\sigma$ from a Rayleigh distribution
Hello stackexchangers,
Suppose we have $n$ Rayleigh distributions defined by
$$f_X(x)=\frac{x}{\sigma^2}e^{-x^2/2\sigma^2}.$$
How would you go about determining an approximative confidence interval ...
2
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1answer
24 views
simplifying an asymptotic expression
I have this expression in a statistics book, namely $nh(f(x) +o(1)+O_p(1/\sqrt{nh}))$. Where $f$ is a density function. Now, this expression is equal to $nhf(x)\{1+o_p(1)\}$. Note, that $n\to ...
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1answer
34 views
How do I prove Poisson appraches Normal distribution
I want to prove why the mean and variance of a $\operatorname{Poisson}(\lambda)$, is different when the time index approaches infinite (it's approximated by the mean and variance of a Normal).
For ...
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2answers
30 views
Show that $-Z$ is also a standard normal random variable.
Show that $-Z$ is also a standard normal random variable; that is, show that $P[-Z < x] = P[Z < x] \,\forall x.$
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1answer
29 views
Is $\left(X_1,… ,X_n,\bar{X}\right)$ jointly normal distributed if $\left(X_1,… ,X_n\right)$ is?
Let $X:=\left(X_1,... ,X_n\right)\sim N_n(\mu,\Sigma)$, $\mu\in\mathbb{R}^n$, $\Sigma\in\mathbb{R}^{n\times n}$ symmetric and positive semi-definite and $\bar{X}:=\frac{1}{n}\sum_{i=1}^n X_i$ as ...
2
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1answer
31 views
How do I calculate typical group size?
If I have a set of groups of individuals (e.g. people), and I want to calculate the typical group size (as observed by individuals), how do I do this?
Wikipedia refers to this as "mean crowding" or ...
2
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1answer
20 views
Derivation of Poisson from Binomial
I am not very well versed in statistics so any clarification would be appreciated. I understand the mathematical derivation of Poisson from Binomial.
I can see just from plotting various Binomial ...
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10 views
Distribution of partial sums of a $L^2$-transformed Gaussian Process
Our assumptions are: $X_t$ is a stationary sequence of standard normal random variables such that $\gamma _X (k)\sim L_{\gamma}(k)k^{2d-1}$ with $d \in (0,1/2)$, where $L_\gamma (k)$ is a slowly ...
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20 views
History of odds making in sports betting
Can anyone provide a reference to the history of odds making in sports betting? In many cases, certain odds are set and then adjusted as people make bets. However, I am having difficulty tracing the ...
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3answers
42 views
How to interpret summation signs
I'm taking a course in statistics, and I really need to brush up my math to be able to follow the book at times.
I'm looking at formulas for sum of squares, and I am slightly confused about the ...
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1answer
15 views
Confidence Intervals under different conditions of Variance??
What's the difference in CI under the situation in part (a) and (b)??
A sample of 20 random streamflow observations produced a mean of 145m3/sec and a variance of 30 (m3/sec)2. What are the 95% ...
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1answer
21 views
First order Autoregressive model
How do I solve this? How can I obtain the lag-one autocorrelation coefficients just from the data??
Following are $10$ years of observation of annual streamflows in millions of cubic meters:
...
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1answer
25 views
Difference between Rician distribution and Gaussian distribution
could any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated
Thank ...
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0answers
16 views
Maximum likelihood estimation of a logistic model
Given $$\mathrm{P}(Z=z/X=x)=\exp{(a+Bx)^t(z)+z^tMz-D(a+Bx,m)}$$ where $$D(a+Bx,m)=\sum_{z \in \Omega} \exp{(a+Bx)^t(z)+z^tMz},$$ $z$ takes $0$ and $1$, and $(z,m,a,b)$ is an $n$-tuple vector and $x$ ...
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23 views
Convolution of logistic function and gaussian distribution
I am trying to solve the folowing problem:
$$\int \exp\left(-\frac{(x-u)^2}{2\sigma^2}\right) \log(1+\exp(ax + b)) \,dx$$
which I think is very complicated and there is no closed form solution(?)
...
