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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Analysis of IQ scores given mean, median, sd, quartiles

The statistics below provides a summary of IQ scores of 100 children Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110 About 50 of the children in this sample have ...
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What is the variation $Var(\hat{y} - y)$ in a linear regression model

There are $N$ data points of the form $(x_i,y_i)$ that corresponds to the model $y_i=kx_i+b+\varepsilon_i$ with $E(\varepsilon)=0$. Coefficients $k$ and $b$ are unknown but we fill the model (all ...
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16 views

Confidence interval - No sample

"According to thorough measurements, the average waiting time at a restaurant is normally distributed with an average of $28$ minutes and a standard deviation of $5$ minutes. Calculate a $95\%$ ...
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24 views

Normal Distribution $E(X^4)$?

So I have the Normal Distribution $f(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}.$ I know any $E(Z^{\mbox{ (any odd #)}})$ makes you integrate an odd function thus giving an answer of zero (i.e. $E(Z^1)$ and ...
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11 views

$E[X^2]$ of the Beta Distribution

So I know the Beta distribution is $$f(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\cdot x^{a-1}\cdot(1-x)^{b-1}$$ I know the $E[X^r] = \dfrac{\Gamma(a+b)\Gamma(a+r)}{\Gamma(a)\Gamma(a+r+b)}$ And I ...
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11 views

Sufficiency of $\bar{X}$ in a Binomial$(2,p)$ population.

Let $\{X_1,X_2,...,X_n\}$ be a random sample from a $\text{Bin}(2,p)$ population. Use the definition of sufficiency to show that the sample mean is sufficient for $p$. Here I am not allowed to use ...
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12 views

Correlation between 3d images and their slices

I work in the field of the image processing and I need to compare results of my algorithm with a gold standart results. For this purpose I calculate the Pearson correlation coefficient between the ...
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30 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
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5 views

Biostatistics book

Do you know a book in biostatistics that fits very well to these topics: Analysis of counts: estimating probabilities, re-expression probabilities as ratios, odds, logits, probits, etc., comparing ...
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30 views

If the variance is $0$ is it constant?

We know that the variance of a constant is $0$. Is the converse also true? Can we say that if the variance of some random variable is $0$ it is a constant?
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What is the confidence level of a curve fitted moving average forecasting next element in time series?

Suppose I got a time series X={X1,X2,X3,...,Xn}, and i got a Moving Average of any kind, it doesn't matter. The moving average is curve fitted to the timeseries of n elements, to fit exactly and most ...
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SOLVE FOR $\sum (X- \overline X )^2$ when $n =1801$ and $S^2= 10$. [on hold]

I know I am definitely missing something. Any help with this would be great!
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4answers
38 views

Expected number of rolls on a dice?

You roll a die until you have seen a 5 on 4 of the rolls (e.g. ⟨5,3,2,5,4,1,6,5,2,5⟩. What is the expected number of rolls this will take? I think that I am way overthinking how I should be going ...
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9 views

Prove SSE=SSPE+SSLF [on hold]

Prove SSE=SSPE+SSLF I can not very understand this statistic equation and its meaning, so that I can not how to prove.
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1answer
15 views

Statistics Questions?!?!

Suppose a relationship between two variables is found to be statistically significant. Explain whether each of the following is true in that case: A) there is definitely a relationship between the ...
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9 views

Finding a confidence interval given sample size, mean, and standard error

I am not the best with Statistics, and I was wondering if it is possible (and how, if it is) to find a 93% Confidence Interval given Sample Size (27), Mean (6.73), and Standard Error (1.732). Thank ...
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20 views

Likelihood interval

Assume that $Y_1, Y_2, \ldots , Y_n$ are independent Poisson random variables each with rate parameter $\lambda$. Assume $n$ is sufficiently large enough so that the central limit theorem (CLT) ...
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40 views

Grad-level math courses to take for career in finance or data science/stats?

I'm a MS student in computational science and want to work in big data/statistics, quantitative finance/HFT, or scientific programming/numerical modeling afterwards. I am currently using Unix, Linux ...
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6 views

How to generate a random vector with a triangular autocorrelation

I am trying to generate N random vectors that when correlated with themselves converge to a triangular function. The approaches I have been trying to follow are reversing a PCA procedure ...
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16 views

Poker Hand Probabilities in Doomtown Reloaded

The short version of my question is as follows: What is the best way to compute the probability of different hand ranks after redraw in poker? Can this easily generalise to a non-standard deck? ...
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21 views

what do these odds ratios represent?

I am reading this article in which is given the matrix of the joint probabilities of two random variables, X=$(x_1,x_2)$ and Y=$(y_1,y_2)$. Let's say they are $(p_{1,1},p_{1,2},p_{2,1},p_{2,2})$. ...
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27 views

Finding the mean of a Maxwell Distribution

Let the velocity of gas particles be modeled by the Maxwell distribution. The probability density function is $$ f(v) = 4\pi \cdot\left( \frac{m}{2\pi K T} \right) ^ {\frac{3}{2}}v^2\cdot ...
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19 views

Confidence interval

I have this statement: If a 95% confidence interval for the mean was computed as (25,50), then if several more samples were taken with the same sample size, then 95% of them would have a ...
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12 views

Find the bias and MSE of an estimator for the upper limit of a uniform distribution.

Let $X_1,\ldots, X_n \sim U(0,θ)$ and let $$\hat{θ} = 2\cdot\dfrac{1}{n}\sum_{i=1}^{n}X_i$$ Find the bias, se, and mse of this estimator. I saw a similar question, but I can't figure out how to get ...
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10 views

Distribution of sample means for large number of samples

normally distributed body weights with a mean of 195 lbs. and a standard deviation of 24 lbs. If an elevator in town has a maximum capacity of 20 persons and a rated maximum weight of 4100 lbs., what ...
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31 views

How to prove a statistic is not complete

Suppose X is a Poisson($\lambda$), where $\lambda\in\{0,1,2,...\}$, how to prove X is not complete. It seems like that we need to find a function $g$ which is not identical $0$, such that the ...
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28 views

The best way to graph a lot of data

I have a lot of feature vectors in the form of: v1=[x0, x1, x2, x3, x4] where x0, x1, and x2 can take binary values. either 0 or 1 x3 and x4 can take values from 0 up to 9 I have a lot of vectors ...
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13 views

Probability that two points (any where on the curve) are a set number of standard deviations apart on a normal distribution

So, here is the question: You buy two pieces of pipe from supplier A, and the inner diameter has a normal distribution of N(muA, sigmaA^2) = N(8.02, 0.1^2). You want these two pipes to butt together ...
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19 views

How do you use the standard normal curve tables?

I was trying to work over a practice problem, and I know I am supposed to use the tables to solve this problem. But I have no idea which part to use where. The question and the tables are given as ...
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Matching Gamma Statistics with Poison Statistics

Confidence intervals with Poisson distribution would be greatly helped by matching an equivalent gamma distribution. Can someone lay out how to match a Gamma Distribution to a poisson distribution? ...
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Multivariate Regression

Suppose there are $n$ variables that map through a function to a single output variable $r$. Given a set of 50-100 data sets with accepted input and output values that satisfy this relation, is it ...
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Equality of two binomial parameters

I am having some trouble with this problem of Ross. Can anyone please help me out. In a famous experiment to determine the efficacy of aspirin in preventing heart attacks, 22,000 healthy middle-aged ...
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Standard deviation of absolute distance of a 1D random walk

Given a 1D random walk (simple +1, -1 movements from the axis) I've seen proofs that the expected absolute distance tends to Sqrt(2*n/PI) and I've plotted graphs of 1D random walks along with this ...
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How to find a probility that the sample mean of a population lies in a particular range?

Assume that X is a random variable with mean x_mean and standard deviation x_sd. If we take a sample of ...
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29 views

Pooled sample variance, how to prove

I did read the related question, and if it did contain the answer to my question, it must have been above my level. This is my very first post, so I'll stick to letters for now. The pooled sample ...
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Calculating uncertainties before aproximating exponential function

I have a table of values that are supposed to be used as points for exponential regression (least squares method). Values in both columns are uncertain, which means that the function itself is a ...
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Calculate total revenue share

Suppose you have data such as this: January Total minutes of all videos watched: 50 Total minutes of video X watched: 25 Total revenue: 200 February Total minutes of all videos watched: 200 Total ...
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A distribution that is skewed to the left [on hold]

Suppose that you and your friends emptied your pockets of coins and recorded the year marked on each coin. The distribution of fates would be skewed to the left. Explain why.
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What percentage of all stocks lost money?

A return less than zero means that owners of the stock lost money. About what percentage of all stocks lost money?
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20 views

Sketch line graphs

Sketch line graphs of a series of observations over time having each of the following characteristics. Mark your time axis in years. (a) A strong downward trend, but no seasonal variation (b) ...
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17 views

Given a joint PDF verify that it is a joint density function

The function from f(x,y) is represented below where 0 < x < 1 and 0 < y < 2 ...
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9 views

Multiple dependant statistical probabilities

How do I calculate probabilities based on each other, for example. If var1 has a probability of 47% , var2 has a probability of 36% if var1 is true, and 0% if false, and var3 has a probability of 23% ...
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asymptotic unbiasedness of weibull mle

It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, ...
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37 views

Confidence Interval has no relation to the probability?

An Intro to Stats class has the following problem: Find and interpret the 90% confidence interval for the true mean The provided answer is this: ...
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5 views

Kruskal Wallis Test Ranking

I have a simple question. I have data which contains one zero and I'm about to rank all the data using excel. Do I need to ignore this one zero value? So that I need to subtract 1 from the entire ...
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38 views

$X$ normally distributed in $\mathbb R^n$ iff components $x_i$ normally distributed?

We've had the normal distribution today in class and I was thinking about the following: Let $X$ be normally distributed, $X\sim N(a,\Sigma)$ with a symmetric positive definite matrix $\Sigma$ and ...
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13 views

Finding Multiple regression coefficients

If I have a multiple regression like this $Y=a+b_1.X_1+b_2.X_2,$ how can I calculate the values of $b_1$ and $b_2$? I have searched on the web but couldn't find an answer.
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Uniform distribution over a set R in the x-y plane and let P be the family of distributions obtained by letting R range over a class T of sets R

So the question is, Let T be the set of all triangles with sides parallel to the x-axis, the y-axis and the line y = x respectively. Find a minimal sufficient statistic and prove that it is minimal ...
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15 views

How to correctly calculate and represent change/delta?

I am a software engineer, and I am generating a report of changes in numbers. I would like to know the correct/standard way of calculating, representing, and displaying the change/delta. For example, ...
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Multiple regression equation

Can someone please recommend me a good book to understand multiple regression? I need to know how to calculate the equation of multiple regression with two independent variables The one I have does ...