Archive for Michel de L’Hôpital

signed mixtures [X’ed]

Posted in Books, Kids, Statistics with tags , , , , , , , , on March 26, 2023 by xi'an

Following a question on X validated, the hypoexponential distribution, I came across (for the second time) a realistic example of a mixture (of exponentials) whose density wrote as a signed mixture, i.e. involving both negative and positive weights (with sum still equal to one). Namely,

\displaystyle f(x)=\sum_i^d \lambda_i e^{-\lambda_ix}\prod_{j=1,i\neq j}^{d}\frac{\lambda_j}{\lambda_j-\lambda_i}\quad x,\lambda_j>0

representing the density of a sum of d Exponential variates. The above is only well-defined when all rates differ, while a more generic definition involving matrix exponentiation exists. But the case when (only) two rates are equal can rather straightforwardly be derived by a direct application of L’Hospital rule, which my friend George considered as the number one calculus rule!

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order, order!

Posted in Books, pictures, Statistics, University life with tags , , , , , , on June 9, 2020 by xi'an

A very standard (one-line) question on X validated, namely whether min(X,Y) could enjoy a finite mean when both X and Y had infinite means [the answer is yes, possibly!] brought a lot of traffic, including an incorrect answer and bringing it to be one of the “Hot Network Questions“, for no clear reason. Beside my half-Cauchy example, some answers pointed out the connection between mean and cdf, as integrated cdf on the negative half-line and integrated complement cdf on the positive half-line, and between mean and quantile function, as

\mathbb E[T(X)]=\int_0^1 T(Q_X(u))\text{d}u

since it nicely expands to

\mathbb E[T(X_{(k)})]=\int_0^1 \frac{u^{k-1}(1-u)^{n-k-1}}{B(k,n-k)}T(Q_X(u))\text{d}u

but I remain bemused by the excitement..! (Including the many answers and the lack of involvement of the OP.)

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