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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33 views

Huber's distribution to minimize the Fisher information (generalize to multivariate case?)

The question is about Robust Statistics (by P.Huber). Any suggestion will be appreciated, thanks. It is proved in Huber's book (Robust Statistics) that the optimal solution of the problem ...
3
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1answer
899 views

How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise.

As the title says, I want to estimate the variances needed for a Kalman filter from real sensor measurements only. For example we can take a temperature sensor, but the solution shall be as ...
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0answers
22 views

How to change 1D Metropolis into 2D?

I've written a MATLAB function to generate random numbers from a given univariate distribution using the Metorpolis algorithm. Here it is: ...
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0answers
19 views

Using the $\chi ^2$ goodness of fit test to see if data is normal.

I have a data sample of size $n$ that I suspect comes from a normal distribution with some parameters $\mu, \sigma$. I wish to check this hypothesis using a $\chi ^2$ g.o.f. test with, say, $\alpha = ...
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1answer
1k views

Expected value of lognormal distribution.

Hi I'm stuck on this question: Recall that X is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if log(X) is normal with mean $\mu$ and variance $\sigma^2$. Suppose X is ...
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1answer
171 views

Dice rolling statistics

So say I roll a 20 sided dice 4 time, the chance of the first roll hitting 1 is 1/20 (5%) but what is the chance of one of the 4 dices coming up with a 1 (and 2 dice coming up with a 1, and 3 dice ...
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2answers
101 views

Sum over product of two binomial distributions

The problem is that of a two-stage "binomial experiment", where first a number $k$ out of $n$ is drawn (each element with probability $p_1$) and later a number $m$ out of those $k$ is drawn (each ...
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2answers
576 views

Difference between two independent binomial random variables with equal success probability

Let $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$ be two independent random variables. Find the distribution of $Z=X-Y$. see also Difference of two binomial random variables I figured this out: $$ ...
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1answer
29 views

How to estimate the variance of several populations when every population mean and variance is different?

I'm currently using the Pooled Variance method, but in my case I cannot assume that every population variance is the same. Is there a method for these cases ?
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0answers
40 views

Stuck on an integral of form $\int\exp(-\frac{\alpha}{m^2} - \beta m)\frac{dm}{m}$. Any ideas?

My statistical model involves the multiplication of a scalar random variable $X|X \geq 0 \sim 2\mathcal{N}(x;0,\sigma^2)\ \mathbb{I} \ [x \in \mathcal{R}_+]$, or a gaussian variable that must be ...
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1answer
40 views

double checking my answers with Probability for or against

1.Find the odds against an event E when pr(E) = 5/6 2.Find the probability of an event when the odds for the event are 6:4 this is what I got for my answer but I am not sure 5/1 3/5
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1answer
43 views

Binomial distribution, when variable isn't x

I've been using the formula $$p(x,N)=\frac{N!}{(\frac{N+x}{2})!(\frac{N-x}{2})!} p^{1/2(N+x)} q^{1/2(N-x)}$$ to determine the probability for a dog who walks in a straight line and can either move ...
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0answers
34 views

How can ANOVA be intuitively understood?

While I have read about ANOVA testing and gone through the formulas in my statistics book, it is not clear to me how the method is motivated or how it has been created. From my understanding, the core ...
2
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0answers
60 views

Notation $x^n=(x_1,\dotsc,x_n)$

In a book on statistics I saw the notation $x^n=(x_1,\dotsc,x_n)$ and wondering how common this is in measure theory/statistics. More precisely it is about a probability space ...
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25 views

normal approximation to poisson

Cotton yarn is wound onto bobbins, each of which takes $100$m of yarn. If the thread breaks before $100$m is reached, the bobbin is rejected. In a trial of a new spinning machine, $13$ bobbins out of ...
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30 views

$100(1-\alpha)$% confidence interval

$100(1-\alpha)$% confidence interval for population mean $\bar Y$ $$\bar y\pm Z_{\frac{\alpha}{2}}\sqrt{\mathbb v(\bar y)}$$ Why is this $Z$ value depending on $\alpha$ important for constructing ...
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1answer
43 views

How to proof that least square estimator $\hat{B}$ doesnt exist when $x$ is linearly dependent?

For the linear regression model $Y=xB+e$, prove that if the columns of $X$ are linearly dependent, the least square estimator $\hat{B}$ does not exist I know that since $\hat{B}$ is an unbiased ...
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1answer
47 views

Exponential(1) distribution of Normally distributed X and Y

Let $X_1,X_2,X_3,X_4,X_5$ be a random sample from the uniform pdf: $f(x)= 1$, $0<x<1$ zero otherwise. Show that $\ln X_i$ has Exponential($1$) distribution for $i=1,2,3,4,5$. Solution: Let ...
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1answer
23 views

Mean and Variance of Nornally distributed distribution

Given X and Y be jointly normally distributed with $\mu_x=20, \mu_Y=40,\sigma_x=3, \sigma_Y=2$ and $\rho=0.6$. Find the mean and the variance of U=X+Y. soln: $U~N(\mu=60,\sigma^2=13). Am I right?$
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31 views

questions about distribution of multivariate normal

I'm looking at this past exam question, For A) Cbhat~N(CU,C(summation)C') B)I have very faint idea of what to do, I tried finding some theroems about distribution but couldn't find any that ...
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1answer
59 views

Maximum Likelihood Estimation with a Gamma distribution

I have this problem that I stumbled upon. Suppose the random variable $X$ follows a Gamma distribution with parameters $\alpha$ and $\beta$ with the probability density function for $x>0$ as ...
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1answer
28 views

proof that a density proportional to Gaussian is Gaussian

I try to develop bayesian estimation for one dimensional Gaussian with unknown $\mu$ and known $\sigma$. I got $$p(x\mid D) = \int p(x\mid\mu)p(\mu\mid D) \, d\mu =\int \frac{1}{\sigma ...
3
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1answer
100 views

How can I recursively approximate a moving average and standard deviation?

Consider a sequence of measurements $(x_1, x_2, ...)$. Let $\mu_n$ be the $p$-period moving average defined by $$\mu_n = \frac{1}{p}\sum_{i=n-p+1}^nx_i$$ and $\sigma_n$ be the $p$-period moving ...
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2answers
31 views

The probability of modulo a prime

Suppose i have a uniform random number generator which generates integers uniformly over some range [x,y] The output obtained z, can be binned into p buckets via: z mod p if p were prime, are the ...
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1answer
20 views

formula for desired sample size

The following is a description of the standard deviation of the sample mean. $$\sigma(\bar X)=\sqrt {V(\bar X)}=\sqrt{\frac {N-n}{N-1}*\frac {\sigma^2}{n}}\leq D$$ Where $D$ is "a constant which ...
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1answer
21 views

confidence intervals for expected spending

A random sample of 10 motorists buying petrol are found to spend an average of £58.30 with estimated standard error £5.25.  Calculate a 95% confidence interval for the expected spending of motorists ...
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10 views

Calculate probability distribution $p\left(\left.X_{1:T}\right|Z_{1:T},y_{1:T}\right)$ in linear- non-Gaussian state space model.

I have a linear, non-Gaussian state space model. Observation equation: $y_{t}=a+bX_{t}+cZ_{t}+\epsilon_{t}$ $\,\,\,\,$ $\epsilon_{t}\sim\mathcal{N}\left(0,\omega^{2}\right)$ Transition equations: ...
2
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0answers
96 views

Accuracy of distance and bearing between GPS locations

I'm writing on an Android app that tracks the distance and bearing between two GPS location (each from a different device). Finding the mean distance and angle between the devices is quite easy, and ...
3
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1answer
75 views

$55$ of registered voters favor incumbent mayor. Find probability that the race ends in a tie.

Fifty-five percent of the registered voters in Sheridanville favor their incumbent mayor in her bid for re-election. If four hundred voters go to the polls, approximate the probability that: (a) the ...
2
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1answer
203 views

chi-square test for uniform distribution

So, i have a hash function which maps a set of possible inputs to a defined range of outputs. I want to test if the mapped outputs are uniformly distributed over the defined range. Wikipedia seems to ...
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1answer
45 views

Linear regression involving angles in a triangle.

In a survey experiment, three independent measurements $29.5^{\circ}$, $30.5^{\circ}$, $120.5^{\circ}$ are obtained from the three angles $\alpha,\beta,\gamma$ of a triangle. Formulate the ...
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27 views

Best metric to measure histogram quality

I'm working on a script that generates histograms for images. It first takes a histogram of all pixels, and then tries various distributions of sample pixels, such as - a grid of pixels, horizontal ...
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1answer
34 views

Joint PDF of two independent normal distributions

I had this on my Probability final, and it stumped me. Exam is over and I still got a B, but here's the problem: Let X1, X2 be distributed as N(0,1) and N(0,9), respectively. Let Y1 = X1-X2, and let ...
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1answer
36 views

Question on regression

So I've been given this formula For regression $R^2=1 - \sum \frac{{(y_i - \hat{y}_i)}^2}{(y_1-\bar{y})^2}$ Now an obvious question that has come to me is why $R^2$ stays the same in certain ...
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0answers
17 views

Is there a name for expressions that are invariant under the exchange of raw moments and cumulants?

I'm interested in expressions that are invariant under the exchange of raw moments and cumulants. This is trivially true of all expressions written only in terms of first order moments but nontrivial ...
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21 views

Trying to find similarity between collection of points

This is a kind of weird problem, and I'm not sure what the best Stack Exchange to post this on is, but I assume Mathematics could help the most. I have many sets of points in 3D space (xyz ...
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0answers
128 views

Proof of mean for a log normal distribution

I've got the following integral $$\int \frac{1}{\sqrt{2\pi \sigma^2}}\mathrm{exp}\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \mathrm{d}x$$ And I'd like rewrite this into something involving an ...
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0answers
58 views

QQ-Plot on R for censored data.

I have plotted a qq-plot on R for a data set which is assigned the value -0.25 for any values below -0.25. However on the qq plot it plots all the -0.25 points, I have been told there is a way to plot ...
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0answers
34 views

Maximum Likelihood Estitmation of a gamma distrbution

I have this problem that I stumbled upon. Suppose the random variable X follows a Gamma distribution with parameters α and β with the probability density function for x>0 as f(x)= [(β^α)/Γ(α)]* x^a-1 ...
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0answers
15 views

To see if data is normally distributed

If I've got some data, and I calculate the mean and standard deviation, and then compare a histogram of said data with the density function related to the parameters from our data.... what would it ...
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0answers
26 views

I'm confused on how to use chi squared for the correlation between age and reaction time

I am doing my IB maths internal assessment and I am confused on how to specifically carry out chi squared with my given data. I will try to explain this quite plainly so the image is clear. I am ...
1
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1answer
34 views

Convergence of sequence averages of rolls of a die to the expected value.

While I was reading about the expected value on the Wikipedia (http://en.wikipedia.org/wiki/Expected_value) one image attracted my attention: Legend of an image: "An illustration of the convergence ...
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2answers
32 views

How to calculate reliabilty of humans in data-entry tasks?

Please help me reason about this: I have many, simple, data-entry tasks to be executed by humans. Of course they will make mistakes, so the result will be wrong sometimes. I don't know, before hand, ...
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2answers
22 views

Finite correction factor, strange version?

My textbook says the following: $$\frac {N-n}{N-1}\cdot\frac{\sigma^2}n \approx \left(1-\frac nN\right)\cdot\frac{\sigma^2}{n}$$ How and Why? This makes no sense whatsoever ^^ Can I prove this ...
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1answer
30 views

Bound for Outlyingness

Given a sample of $n$ data, $x_1, \dots, x_n$. Define the sample mean $$\bar x := \frac{1}{n}(x_1+\cdots+x_n),$$ and sample variance $$s^2 := \frac{1}{n-1} \sum_{i=1}^n (x_i-\bar x)^2.$$ To measure ...
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0answers
24 views

Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
2
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1answer
38 views

Need to know why $\sum_{k=0}^{\infty}kr^{k} = \frac{r}{(1-r)^{2}}$

Working on a Stat problem where I must find $E(x)$ of $f(x)=\left(\frac{1}{2}\right)^{x+1}$ for $x=0,1,2,\cdots$ I have, ...
0
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1answer
58 views

Jayne's Equation 1.13 Derivation

Dear Stack Exchange Members, I'm reading 'Probability Theory - The Logic of of Science" by ET Jaynes, and I'm on pg. 11. Jayne's says: *"...For example, we shall presently have use for a rather ...
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13 views

Covariance and Correlation of multiple random variables

The problem I'm working on is: Let $W,X,Y,Z$ be i.i.d. with common variance $\sigma^2$. I need to find $Cov(W+X+Y,X+Y+Z)$ and $\rho(W+X+Y,X+Y+Z)$. So far I have: $Cov(W+X+Y,X+Y+Z)= Cov(W,X) + ...
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1answer
29 views

Probability density function / maximum likelihood for correlating sequence

I have a stream that contains two consecutive identical sequences, each of length $N$. These sequences have a ideal autocorrelation property. So I want to have the probability density function over ...