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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Can I sum variances to a compound variance?

Say I have three locations A,B,C and I have a person going from A to B and measure the time it takes. Same for B to C. Let the variance of the time it takes for the path AB be a and for BC b. Is it ...
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36 views

adjusted R squared with multiple dependent varialbles

A question about regression in statistics. What is the formula for adjusted R squared if there are multiple dependent variables
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42 views

Estimating the mode

I am interested in the following problem and after searching I am surprised I can't find anything useful about it. Consider a multiset $A= \{a_1,\dots, a_n\}$ of integers. Say you sample elements of ...
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1answer
28 views

Probability Density Function given X

I have this problem I am unable to solve in my book. It has a provided solution and I am unable to come to this conclusion. The problem is as follows: Suppose that a random variable X has a PDF given ...
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20 views

How to calculate p-value using an algorithm on R assuming that distribution is unknown.

For a given sample x, where x is distributed with a normal distribution mean known but variance unknown. I am testing the hypothesis that variance is equal to one or greater than 1. The question is to ...
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19 views

Montmort's card matching problem: Distribution of the number of matching cards?

(Introduction to Probability, Blitzstein and Nwang) Recall de Montmort’s matching problem from Chapter 1: in a deck of n cards labeled 1 through n, a match occurs when the number on the card ...
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Let $X \sim \text{HGeom}(w,b,n)$, what is the distribution of $n-X$?

Let $X \sim \text{HGeom}(w,b,n)$, what is the distribution of $n-X$? The distribution of $X$ (e.g., number of white ($w$) balls in a sample of size $n$) is hypergeometric, so $$P(X=x) = ...
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29 views

Using gamma distribution to find the average duration of breaks after 10 calls with exponential distribution

Worker works 8 hours a day. Time between $ 2$ calls has $\exp(4)$ distrubution (expecting $4$ calls per hour). Duration of calls is $0$ (he just registers them). After $10$ calls he goes to $15$ ...
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23 views

The autocorrelation function - the result in the form of a vector.

I've implemented the autocorrelation function in Python according to the normalized autocovariance function for discrete signals, i.e: ...
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15 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 ...
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9 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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1answer
36 views

Expected value of variance and variance of variance?

From a certain system dynamical system I am able to calculate the expected error $\mathbb{E}(e_k)$ and the variance $\sigma^2 = \operatorname{var}(e_k) = \mathbb{E}(e_k^2) - \mathbb{E}(e_k)^2$, as ...
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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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13 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
17 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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20 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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37 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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21 views

Calculating expectation function and covariance function

Let $E_n(t)$ denote the empirical cdf based on iid uniform $u[0,1]$ random variables $U_1,...,U_n.$ The corresponding uniform empirical process $(e_n(t),0\leq t\leq 1)$ is given by ...
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44 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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15 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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34 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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28 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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31 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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Q: 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 ...
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55 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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21 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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23 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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25 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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14 views

simulate inventory model with antithetic variable [closed]

I need make a simple program of the inventory model with scilab or matlab. I have the code but I need add the antithetic method for reduce the iterations's number with the variance reduction and I ...
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1answer
53 views

Bernoulli and Gamma distribution question [closed]

the probability of your bike breaking down on any given day can be modeled as a Bernoulli random variable with probability of breaking down t. Furthermore, assume that if your bike breaks down the ...
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Mixed distribution question (stats) [closed]

deleted due to duplicate question Suppose that the probability of your bike breaking down on any given day can be modeled as a Bernoulli random variable with probability of breaking down p. ...
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1answer
28 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
16 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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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
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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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Continuous and mixed conditioning

Question removed due to duplicates Consider the following probability model: Z ~ Normal(0,1) Y|Z ~ Normal(Z,1) Show that fZ|Y(z|y) is a normal density. Find the parameters of this updated ...
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1answer
21 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 ...
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1answer
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Suppose you have a set of data. Find a formula for the curve y=ax^3 using least squares [closed]

I am really stuck on this theoretical question, practicing for my final. What are the exact steps needed to get to this point?
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1answer
54 views
+100

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
21 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
16 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
14 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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7 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: ...
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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 ...
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1answer
32 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 ...
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1answer
25 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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13 views

How does a two-factor design allow you to control or reduce error variability? [closed]

How does a two-factor design allow you to control or reduce error variability?
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26 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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7 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
18 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 ...