# whuber

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bio website quantdec.com location Northeastern US age 13 member for 3 years, 3 months seen 25 mins ago profile views 15,088

Consultant (environmental and spatial stats a specialty), expert witness, and teacher. I can be reached through (outdated but still valid) links posted on my web site.

 Nov27 comment Expected Value of cumulative distribution function You seem to use "exist" in the sense of "has a conventional closed-form expression." ("Exist" in math and stats usually means that it has a well-defined value, whether or not that value can easily be written.) This raises the question of what you mean by "compute": are you looking for a closed-form expression, or would alternatives (such as a power series, and algorithm, or a numerical approximation) be acceptable? Nov27 comment R - Find curve pattern of simple time series Is all this really necessary when the O.P. merely requests a set of descriptive statistics? Nov27 comment Correlation of distribution standard deviation with subtraction of its mean from mean of normal distribution It is important to edit your question to reflect these refinements, rather than embed them in comments. The usual senses of "correlation" do not give any meaning to your use of this word: I suspect you may be employing it in a non-standard or informal way. That needs to be explained further. Nov27 comment Variance of sample mean of bootstrap sample You're probably right--but this answer doesn't seem terribly informative. Perhaps you could point to which part is not correct? Nov26 comment Predicting Data. We cannot opine (objectively, anyway) about "appropriate" ways until the question is sufficiently well-defined. At this point practically any answer would satisfy some definition of "best." The crux of the matter is that your assumption "the more units we manufacture the lesser we rest we give to the machines" is not quantitative and (therefore) cannot be used effectively. Nov26 comment Predicting Data. This problem does not appear to have enough information to be be answerable. Is it a real problem that you face? If so, please clarify what you mean by "best." Nov26 comment Is the absolute value of the difference between two Poisson distributions a Poisson distribution? @prob It is a modified Bessel function of the first kind. I exhibit the (simple) derivation for $x=0$ at stats.stackexchange.com/questions/49991/poisson-processes/…; it will proceed similarly for general $x$. Nov26 comment How can the formula for the expectation of a log-normal random variable be dimensionally sound? $\mu$ is a logarithm: what units of measure do you suppose it has? Nov26 comment Gambler's ruin, fun problem Although this answer may be correct, it needs justification. The certainty of an eventual loss does not imply the expected winnings are zero! The standard example of this occurs when a gambler has even odds on a fair coin flip and just lets her winnings ride: although she is certain to lose (in the limit as the number of plays increases), the limiting expected value of this strategy is one-half her initial stake. Nov26 comment Probability distribution for different probabilities @wolfies Yes, that's an interesting idea to carry out. It would not be relevant to my answer here, though, because I do not advocate using the Normal approximation in these cases for two explicitly stated reasons: 16 is not "many more than 16 trials" and in both the sequences $(i/17)$ and $(\sqrt{i}/17)$ some probabilities will "get too small." For the case of $2^{16}$ trials with variable probabilities please see stats.stackexchange.com/a/5482 and for a discussion of other approximations (which aren't very good IMHO) read the remainder of that thread and its comments. Nov25 comment OLS unbiased for all sample sizes Search on "Gauss Markov Theorem." Nov25 comment OLS unbiased for all sample sizes Isn't the Wikipedia article on OLS good enough? Nov25 comment ARIMA estimation by hand 1e-100 when added to the values of sigma^2 which typically would be passed as arguments to your function will be the same as adding $0$ (you only have 16 sig figs or so, not 100), so you have shown you don't need the additive offset. I was cautious in my earlier remark: I did not assert that these results would be affected, but that they would when $\sigma$ is sufficiently small. Nov25 comment Interpreting gstat fit.variogram - partial sill vs sill I have voted to close this as a duplicate because the response I posted there (just a few hours ago!) fully answers your question. Briefly, the sill is the difference between the asymptotic level and the nugget. @Nick Use your Web browser's controls to zoom in: the image is perfectly clear and detailed, but is shrunk by SE to fit within its standard width. Nov25 comment ARIMA estimation by hand If you keep the additive offsets to sigma^2, you will need to solve for sigma after running the optimization (by subtracting the offset from the fitted value and taking the square root of the result). Nov25 comment Expectation of a function of a random variable @Michael True: but the point is that you first have to show that $E[g(X)]$ truly is a distributional property: to assume that it is amounts to a circular argument. (I am not asserting this is anything but obvious and easy to prove, but it does need to be shown.) The law has its memorable name because so many people appear to be unaware of this logical gap in your assertion. Nov25 comment Expectation of a function of a random variable The notation in your previous comment makes no sense: the value of a realization, such as "$0.5$", cannot have an expectation! You seem to be confusing the realizations with the random values used to model them. The value $0.5$ is a realization of a Uniform$[0,1]$ variable; likewise, $0.75$ is a realization of another Uniform$[0,1]$ variable. The variables--which are measurable real-valued functions defined on a sample space--have expectations, while their realizations are just numbers (which we anticipate will change in repetitions of the experiment). Nov25 comment ARIMA estimation by hand What is the relationship between $T$ and $n$? Your code has no "$T$" in it. Perhaps $T=n+1$? If so, there would be an obvious bug in the calculation of g1. You need to dispense with the +0.000000001 stuff, too, even though that changes the answer only a little: for small $\sigma$ it will ensure you report incorrect values of this parameter. Nov25 comment Expectation of a function of a random variable There is a subtlety that the Wikipedia article perhaps does not sufficiently emphasize: two random variables may differ but have the same distribution. For instance, let $X_1$ be the length of a chord of a unit circle whose endpoints are obtained from two independent directions uniformly distributed on $[0,2\pi)$. Let $X_2$ be same thing, but change its value to $-1$ on all diameters of the circle. These variables, although they differ on an infinite set, still have the same distribution. Nov25 comment Expectation of a function of a random variable Independence seems irrelevant here. If $X_i$ and $X_j$ are any random variables with the same distribution, then what needs to be shown is that $E[g(X_i)]=E[g(X_j)]$ for any measurable function $g$ (see "Law of the unconscious statistician"‌​).