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July 2

Infinity and existence

1. Is it conventionally acceptable to write E[X] < ∞ when you mean that E[X] is finite?

2. Is it conventionally acceptable to say that E[X] does not exist when you mean that E[X] = ∞?

Thanks in advance! Loraof (talk) 17:42, 2 July 2017 (UTC)[reply]

I've seen this more commonly applied to integrals (of which E[X] for continuous distributions is a special case) when discussing absolute convergence. One should distinguish between the cases of having infinite value and no value (e.g.

\int _{-\infty }^{\infty }{\frac {x}{1+x^{2}}}dx

does not exist, whereas one could safely write

\int _{-\infty }^{\infty }x^{2}dx=\infty

).--Jasper Deng (talk) 18:10, 2 July 2017 (UTC)[reply]

Convention depends on context, but I would say "yes" to the first and "maybe" to the second (for the reason Jasper Deng mentions). --JBL (talk) 20:48, 2 July 2017 (UTC)[reply]

I'd say "yes" to the first and "no" to the second, but opinions may differ. -- Meni Rosenfeld (talk) 08:52, 3 July 2017 (UTC)[reply]

I agree with Meni. The "does not exist" nomenclature commonly excluded +/- infinity, if not always. This is because things like expected values are often implicitly treated as belonging to the Extended real numbers, and in that case, E[X] d.n.e is distinctly different from E[X]=+\- ∞. SemanticMantis (talk) 16:12, 3 July 2017 (UTC)[reply]

Trigonometric Identities

How many trigonometric identities are there in mathematics?41.58.87.37 (talk) 19:04, 2 July 2017 (UTC)[reply]

I am not sure that there is any upper limit on their number. Ruslik_Zero 19:57, 2 July 2017 (UTC)[reply]

There are infinitely many of them. For example, for every positive integer value of n, there is an identity equating cos(nx) to an nth degree polynomial in cos(x). Loraof (talk) 21:55, 2 July 2017 (UTC)[reply]