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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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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12 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
20 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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22 views

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
22 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
68 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
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
15 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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0answers
14 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
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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1answer
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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0answers
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 ...
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1answer
28 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
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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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17 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
23 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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23 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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21 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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12 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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13 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 ...
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0answers
29 views

demonstration logistics function

I'm using a model of the Logistic Equation. The reasoning is as follows: $dP/dt = K.P$ (Basic population model) To use a more realistic function and limit growth $dP/dt = (A - BP)P$ (A and B ...
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1answer
23 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
18 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
18 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
28 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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21 views

Birthday question regarding 3 people in one family having same birthday [duplicate]

I'm trying to computer the probability of 3 people in the same family having the same birthday. By my calculations, it should be: $$\frac{ 365^3 }{ \text{number of people in the family}}$$ Would I ...
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0answers
15 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 $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
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1answer
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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, ...
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1answer
37 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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7 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
20 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 ...
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1answer
32 views

Applying Bayes Rule to Cards

I was playing poker with a friend last night when a question occured to us. I had a two Jacks and the flop came out: King Queen and 4. So, suddenly my pocket Jacks are not so great, unless another ...
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2answers
26 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0$ + integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.
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1answer
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Show that: $ X \ {\text{is}}\ \mu{\text{-integrable}} \implies \sum_{k=1}^\infty\mu(\{\mid X\mid ≥ k\}) < \infty$

Assignment: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $X: \Omega \rightarrow \bar{\mathbb{R}}$ a $\mathfrak{A}$-$\bar{\mathfrak{B}}$-measurable function. Show that: $$ X \ ...
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2answers
32 views

Show that: $\int_{\Omega} X d\mu = 0 \iff \mu(\{\omega\in\Omega\mid X(w)>0\})=0$

Assignment: Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and $X: \Omega \rightarrow \bar{\mathbb{R}}$ a non-negative $\mathfrak{A}$-$\bar{\mathfrak{B}}$-measurable function. Show that: ...
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1answer
30 views

Show that $\bar g=\frac1 N \sum_{i=1}^Ng(x_i,y_i)=g(\bar x,\bar y)$

I have been asked to show that: $$\frac1 N \sum_{i=1}^Ng(x_i,y_i)=g(\bar x,\bar y)$$ Where $$\bar g=\frac1 N \sum_{i=1}^Ng(x_i,y_i)$$ A clue given in the problem sheet is: $$g(x_i,y_i)-g(\bar x,\bar ...
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1answer
31 views

Recalculate R^2 deleting 1 point

Is there a way to recalculate $R^2$ of a regression that I delete a point (for example an outlier point)? The idea is to get the $R^2$ without a point but without recalculating all the regression. ...
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11 views

Ranking system that factors in both positioning and clustering?

I'm not sure how to express my problem in proper semantic terms, so please forgive me if I am unclear, waffling or use the wrong terminology anywhere here! I'm trying to find a ranking system that ...
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2answers
46 views

Calculate the mean of the normal distribution function $\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ by integration.

I know that it must be $\mu$ but I cannot get the answer. This is my attempt so far: Normal distribution function = $N(x)=\frac1 {2\pi \sigma^2}exp[-\frac {(x-\mu)^2} {2\sigma^2}]$ $$\langle ...
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1answer
22 views

Central sample moments are asymptotically unbiased

Let $\newcommand\top{\overset p\to}\newcommand\isd{\overset d=}\newcommand\P{\mathcal P}\DeclareMathOperator\var{Var}$$X\isd X_1\isd x_2\isd\ldots\isd X_n$ be independent stochastic variables with ...
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3answers
63 views

Show that the variance, $\mathbb E((x-\mathbb E(x))^2)$, can be written as $\mathbb E(x^2)-(\mathbb E(x))^2$

This question has been set in the Christmas work for the chemists at oxford uni and the hint that was given in the problem sheet was "does $\mathbb E(x)$ depend on $x$?". There is a derivation on ...
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15 views

Collection of values: Root mean-square?

I am currently developing a music recommander system using the Million Song Dataset. This dataset contains 1mio songs and around 50 attributes related to them (how hot it is, the tempo, the key, the ...
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1answer
31 views

What does the expectation value of $x$ mean? Surely it must expectation value of a function of $x$?

How can a value of $x$ have an expectation value? Surely there must be a distribution of values of $x$ for the expectation value to be calculated. Is this the reason for the normal distribution ...
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1answer
32 views

Logistics functions and values for pair of points.

I have a question to them: I have the following pairs of points: $(0.005, 1200),(0.015, 3500), (0.02, 4400), (0.025, 5000)$ How I can find the appropriate logistics function? Thank you!