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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1answer
33 views
Find the limiting distribution
Find the limiting distribution for $n\rightarrow \infty \text{ of} \prod\limits^n_{i=1}X_i$. Given is that $f(x)=\frac{1}{2x\sqrt{2\pi}}e^{-\frac{1}{8}(\ln x-\theta)^2}, x\geq 0$.
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1answer
20 views
This is regarding Chi square test
A chi square test is conducted to check whether a person's ability in Mathematics has an impact on his/her interest in Statistic. The test statistic is 13.277 under the tested null hypothesis.
write a ...
1
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1answer
20 views
Simple Linear Regression Question
Let $Y_{i} = \beta_{0} + \beta_{1}X_{i} + \epsilon_{i}$ be a simple linear regression model with independent errors and iid normal distribution. If $X_{i}$ are fixed what is the distribution of ...
4
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1answer
154 views
source of proof for a characterization of normal distribution
I want to know the proof of the following statement about normal distribution:
If the sample mean and sample variance are independent for a population, then the distribution of the population is ...
1
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1answer
21 views
Ratio of PDF to complementary CDF
Let $f(x)$ be a probability density function, and $F(x)$ be the cumulative distribution function of $f(x)$.
$$F(x) = \int_{-\infty}^{x}f(u)du$$
Then intuitively, what does the following ratio ...
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8 views
this is regarding hypothesis testing under chi square test
It is desired to test whether the number of gamma rays emitted per seconf=d by a ccertain radioactive substance is a random variable having th Poisson distribution with λ=2.4
use the data given below ...
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12 views
Multivariate regression
What is the most suitable way to assess the effect several independent ordinal variables have on a single nominal variable ?
For example the chance of a tumour being malignant predicted by the
1. ...
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1answer
27 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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1answer
25 views
Find the Cramer-Rao bound for an unbiased estimator of $b^2$
$X$ is a RV with pdf $f(x,b) = \frac{x}{b^2} \exp \{-\frac{x^2}{2b^2} \}$
I've got two different estimates: $\hat{b^2} =\frac{2}{\pi} (\frac{1}{n} \sum_{i=1}^n X_i)^2 $ using MME, and $\hat{b^2} = ...
0
votes
1answer
124 views
What does relaxing the iid assumptions mean? Intuitive and technical perspectives.
I believe the most restrictive assumption we can place on a series of observations is that they are iid.
It is possible to relax these assumptions. For example relaxing the independent distribution ...
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0answers
25 views
Calculating the probabilities of different lengths of repetitions of numbers of length 6
This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4
except now I'm having problem with numbers of length 6.
...
0
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1answer
14 views
Distribution for Response Times
I have samples from a response time population for a web transaction. I want to be able to use them to describe a distribution for the population but don't know a proper one to use. I have shied away ...
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0answers
16 views
Survival Analysis Partial Likelihood
Part of a medical statistics course is on Survival Analysis. We are introduced to the Cox Proportional Hazard model and then move on to look at partial likelihood $L(\beta)$. There is little on what ...
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1answer
32 views
Calculating the probabilities of different lengths of repetitions of numbers of length 4
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 ...
1
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0answers
17 views
Normalizing the Second Moment of $n$ Discs
Consider $n$ non-overlapping discs of diameter $d$ positioned (centred) at $P_1,\dots,P_n$ ($||P_i - P_j||\geq d, i\neq j$).
Graham and Sloane use the second moment as a measure of compactness for ...
0
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1answer
17 views
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 ...
1
vote
1answer
61 views
How do I divide a set of data samples which follow a logarithmic distribution?
I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the ...
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2answers
42 views
Check for sufficiency
We have the function
$$f(x) = \frac{1}{2x\sqrt{2\pi}}e^{-\frac{(\ln x - \theta)^2}{8}}$$ for the I.I.D (identically independent distributed) sample $X_1,\dots, X_n$.
I have to show that ...
1
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1answer
19 views
Continuous random variable question
$ X $ is a non-negative continuous random variable with density function $f$ and distribution function $F$.
Use integration by parts to show that
$ \int_0^{\infty} ( 1- F(x)) dx =
...
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1answer
288 views
Is standard deviation additive?
I am helping someone study for a statistics exam. I am quite good at most other math classes but it's been a while since I studied statistics. I am stuck on one of the exercise problems that we worked ...
2
votes
1answer
30 views
Range of the distribution of $(1-X)$ when $X$ follows Beta distribution as $X\sim beta(p,q)$
if $X$ follows beta distribution with parameter $p$ and $q$ where $p>0\quad , q>0$
then $1-X$ follows beta distribution with parameters $q$ and $p$,
that is if $X\sim beta(p,q)$ then ...
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0answers
10 views
Time Series: Whether a proces is invertible or not ?
If the roots of a characteristic polynomial of a MA(2) process are on the unit circle, i.e. z = 1,
is the process still invertible?
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1answer
21 views
expected value of random variables
Take two random variables $X=a+bX_0$ and $Y=c+dY_0$, and define $T=X-Y=\mu+\sigma Z$ where $\mu$ is the mean of $T$, $\sigma$ its standard deviation and $Z$ is a standardized random variable with mean ...
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2answers
450 views
How to generate and use random trees?
I have the assignment to implement a random tree classifier in MATLAB.
The lecture says:
...
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0answers
9 views
simple trend measure/score
I am looking for a very simple (potentially ‘parameterisable’) measure to determine the trend of a discrete time series where the measurements are not necessarily equidistant. The output of the ...
0
votes
1answer
149 views
Need help finding value (or contingency table) for Chi-squared critical value.
I need help finding value (or contingency table) for Chi-squared critical value at 95% significance level when degrees of freedom is 58.
I have calculated the chi-square calculated value, and I need ...
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3answers
2k views
Calculating interquartile range
I have the following numbers:
$$\{0, 1, 2, 5, 8, 8, 9, 10, 12, 14, 18, 20, 21, 23, 25, 27, 34, 43\}$$
and need to calculate the IQR. My calculations gave me:
$$18/4=4.5$$
$$Q1=(5+8)/2=6.5$$
...
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votes
0answers
47 views
Product of Uniform and Gamma Random Variables
Let $X\sim\operatorname{Gamma}(1+\alpha,1)$ and $U\sim \operatorname{U}[0,1]$ be independent, $\alpha < 1$
How do you go about proving that $XU^\frac{1}{\alpha}\sim\operatorname{Gamma}(\alpha,1)$?
...
5
votes
1answer
84 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
votes
0answers
82 views
tolerance intervals, solving for probability
Given a small sample from a normally-distributed population, how do I calculate the probability that a specified percentage of the population that is within some bounds [A,B]?
I'm trying to do some ...
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1answer
22 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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votes
1answer
18 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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votes
1answer
49 views
UMVUE of λ in Exp(λ)
Suppose (X1,...,Xn) is a sample from a Exp(λ) population. Try to find an UMVUE of λ. Remember S = Sum[Xi,i=1,n] is both complete and sufficient for the Exponential Family.
I'm trying to use the ...
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vote
3answers
126 views
What is a direct correlation?
I have two contrary definitions of for the direct correlation between two variables $X$ and $Y$
Their correlation coefficient is close to $1$.
There is a direct causal relationship between the ...
0
votes
1answer
590 views
Uniform density question
If X1...Xn are i.i.d. uniform(0, theta), x>0 and x<=theta, and theta>0, Calculate t(theta) = E(X) and find the BUE.
a. Find t(theta) = E(X|theta) and find the best unbiased estimator
Attempt: ...
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1answer
172 views
Balls in a box probabilities
A box is filled out by 1000 balls. the box can be thought as containing V sites and V balls, with V=1000. The box is repeatedly shaken, so that each ball has enough time to visit all 1000 sites. The ...
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0answers
42 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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0answers
42 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$. ...
2
votes
0answers
22 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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0answers
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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vote
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 ...
9
votes
2answers
258 views
Fast computation/estimation of the nuclear norm of a matrix
The nuclear norm of a matrix is defined as the sum of its singular values, as given by the Singular Value Decomposition of the matrix itself. It is of central importance in Signal Processing and ...
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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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2answers
30 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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1answer
111 views
+50
How does one prove probability integral transform?
How does one prove probability integral transform? So when $Y = F_X(X)$ where $X$ has a continuous distribution for which the cumulative distribution function is $F_X$, why does $Y$ have a uniform ...
2
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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 ...
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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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0answers
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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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:
$$
...
