Dennis
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Discrete analog of CDF: "cumulative mass function"?
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17 votes

The proper terminology is Cumulative Distribution Function, (CDF). The CDF is defined as $$F_X(x) = \mathrm{P}\{X \leq x\}.$$ With this definition, the nature of the random variable $X$ is irrelevant:...

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History: the role of statistics in astronomy
13 votes

Probably the best-known example of a statistical method "developed" from an astronomy problem was Gauss' use of least squares to generate an orbit for Ceres on the basis of Piazzi's observations. ...

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Do the pdf and the pmf and the cdf contain the same information?
11 votes

PMFs are associated with discrete random variables, PDFs with continuous random variables. For any type of random of random variable, the CDF always exists (and is unique), defined as $$F_X(x) = P\{X ...

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What is a block in experimental design?
6 votes

Experimental designs are a combination of three structures: The treatment structure: How are treatments formed from factors of interest? The design structure: How are experimental units grouped and ...

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Proper analyses for 2x2 contingency tables
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5 votes

Pearson's $\chi^2$ test is useful for a sample of $n$ observations cross-classified by two variables, say $A$ and $B$. These tests test the null hypothesis that $A$ and $B$ are independent variables. ...

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Are integer results from random number generators unlikely?
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5 votes

Let's start from theory, and worry about "How To" later. Let's suppose that $U \sim \mathrm{Uniform}(0, 10)$. Now, fix an integer, say $i$. The probability that $i-\epsilon/2 < U < i+\epsilon/...

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What subject is more useful for statistics students
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5 votes

If those are the choices, then Complex Analysis is probably the way to go. PDEs are interesting and useful, but not too much in statistics. Complex Analysis, on the other hand does pop up from time ...

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What is a "factor" in factor analysis?
4 votes

The usual factor analysis model is $$\mathbf{Y} = \mathbf{\mu}+ \mathbf{\Phi}\mathbf{L} + \mathbf{\eta},$$ where $\mathbf{Y}$ represents a collection of $n$ observations of $k$ random variables; i....

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Outliers and the mean
4 votes

I don't know that it's called anything in Mathematics. In Statistics, it's an example of a trimmed mean, and particularly a 14% trimmed mean.

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When is Maximum Likelihood the same as Least Squares
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4 votes

Levenberg-Marquardt is a general (nonlinear) optimization technique. It is not specific to LS, although that is probably its widest use. Looking at your referenced paper, they are (mostly) fitting ...

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relationship between ARMA and AR
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4 votes

As I'm sure you are aware, ARMA is an acronym for AutoRegressive Moving Average (Stochastic Process). More fully, we use ARMA(p, q) where p is the order of the autoregressive portion and q the order ...

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Should I use t-test on highly skewed and discrete data?
3 votes

To $T$ or not to $T$ -- is that the question? I would suggest backing off for a moment and asking yourself, "What IS the question?" Is the question, "Are the means of populations 1 and 2 the same?", ...

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How can I create a testable hypotheses to compare benchmark results for the same tests run in 3 environments
3 votes

Don't do hypothesis testing. It's only going to tell you what you already know: they are different. Quantify the differences in your response variable(s) across the environment. You can do it in a ...

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Excel: which products are most frequently ordered together? (clustering question)
3 votes

Friends don't let friends use Excel for serious statistics. Yes, that is a slam on the folks in Redmond. When they fix their pseudorandom number generator I'll take another look at Excel. Now that ...

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Probability of event happening when data is aggregated with many independent events over the course of time
2 votes

Starting with the case of two coins, we know that $$E\{T_k \}=m_k\theta_A+n_k\theta_B.$$ If $k=2$, we can solve $$t_1 = m_1\hat{\theta}_A +n_1\hat{\theta}_B$$ $$t_2 = m_2\hat{\theta}_A +n_2\hat{\theta}...

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Obtaining an estimator via Rao-Blackwell theorem
2 votes

The fact is that Alecos' answer is the easiest way to handle the problem, but the problem can be solved via Rao-Blackwell as well. Start with the joint density $$f(x_1,..., x_n | \theta) = e^{-\sum ...

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What is the maximum likelihood estimate of the covariance of bivariate normal data when mean and variance are known?
2 votes

Under the stated conditions ($\mu_X = \mu_Y = 0$ and $\sigma_X = \sigma_Y = 1$), the likelihood function for a random sample of size $n$ is $$L(\rho\; |\; X, Y) = \frac{1}{(2\pi[1-\rho^2])^{n/2}}\exp \...

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Consistency of unbiased estimator of error term variance in Multiple regression
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2 votes

You know that $S^2_e = \frac{1}{n-k}\mathbf{e}'\mathbf{e}$ is an unbiased estimator of $\sigma^2$. So, if you show that $Var(S^2_e) \to 0$ as $n \to \infty$ you've shown that $S^2_e$ is consistent (i....

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Asymptotic distribution of uniform order statistics
2 votes

I'm going to assume this is self-study, and make some suggestions rather than giving an explicit answer. The proof is in two steps: Find the joint distribution function of $U_{(1)}, U_{(n)}$; Show ...

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Confidence Interval and Signifcance test Question
2 votes

The alternative hypothesis is $H_1: \mu \neq \,\$95,000.$ The significance level is the same as the error rate of the confidence interval, $\alpha = 0.01$. You cannot recover an observed ...

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Can I do a t-test Power Analysis for Unequal Size Groups which Produces 2 Different Minimum n's?
2 votes

First off, why are you assuming equal variances in the two groups? Please don't say, "Because it's convenient." I seriously doubt that the group variances are equal, although in the case of equal ...

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Imputation and Distributions
2 votes

It depends on what you are trying to accomplish. What distribution do you want the imputation to reflect? $$\mathrm{N}(10, 25)$$ or $$\mathrm{N}(10,\frac{25}{\sqrt{1000}})?$$ Your second task could ...

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Using more measurements of lower quality or just the one with the best quality?
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2 votes

It is pretty well known that $$\mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2 \mathrm{Cov}(X,Y)$$ or in terms of $\rho$, $\sigma^2_X$ and $\sigma^2_Y$ $$\sigma^2_{X+Y} = \sigma^2_X + \...

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How to show linear model corresponds to exponential family?
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2 votes

Nope. What happened to your product term, $\sum_{i,j}x^{(i)}_j\cdot y_i$? You'll find this clearer if you write it in matrix notation rather than scalar notation. Your joint density is then $f(\...

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Data Acquisition in R
2 votes

You don't need anything but base R. Use the scan( ) function and pipe from LabVIEW to stdio.

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Controled Factorial Experiment Design: which name or resource?
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2 votes

It is a repeated-measures factorial design, albeit of a very simple sort. You have one "within-subject" factor (pre-test and post-test (after treatment) and two "between-subjects" factors (A and B). ...

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Sample size vs response rate, which is more important?
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2 votes

You are assuming that non-respondent data are missing at random (MAR). MAR is a very strong assumption, and not generally borne out in practice. You already know that respondents and non-respondents ...

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Relation between mean of the hypergeometric distribution and binomial
2 votes

The mean of the hypergeometric distribution can be interpreted as the finite sampling equivalent of $\mu = np$ from the binomial, taking $p=\frac{K}{N}$. The variance can be expressed as $\sigma^2 = \...

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Independence of a linear and a quadratic form
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2 votes

Use Craig's Theorem. Consider the quadratic form on b. If two random variables are independent, then any univariate functions of those random variables are likewise independent. The quadratic forms ...

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Correlation coefficient: If $\rho = 0$, then $r$ is normally distributed with mean 0. Why?
Accepted answer
2 votes

Oy veh iz mir... The source is wrong. In fact, the source acknowledges that it is wrong immediately below the claim when it states that under $H_o:\rho=0$ and when $(X, Y)$ are jointly Normal, $T = ...

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