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January 8

Overlapping deviations

When you have averaged measurements for two items, what is the probability the one with the larger mean is actually the smaller, assuming normal distributions? That is, what is the chance that green is greater than yellow here? (without requiring the σ's to be equal). — kwami (talk) 01:41, 8 January 2014 (UTC)[reply]

I'm not 100% sure what you're trying to ask, but if you're asking, given normally distributed variables

X

and

Y

what is the probability

X<Y

, it is the same as the probability that

X-Y<0

. If

X

and

Y

are normally distributed, then

X-Y

is a normally distributed random variable with mean

\mu _{X-Y}=\mu _{X}-\mu _{Y}

and variance

\sigma _{X-Y}^{2}=\sigma _{X}^{2}+\sigma _{Y}^{2}

. The probability that this variable is less than zero is just by definition the cdf of the normal distribution with these parameters evaluated at zero; that is,

\Phi \left({\frac {\mu _{Y}-\mu _{X}}{\sqrt {\sigma _{X}^{2}+\sigma _{Y}^{2}}}}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\mu _{Y}-\mu _{X}}{{\sqrt {\sigma _{X}^{2}+\sigma _{Y}^{2}}}{\sqrt {2}}}}\right)\right]

. These functions have no closed form and you usually look their values up in a table. 130.56.71.53 (talk) 06:53, 8 January 2014 (UTC)[reply]

Thank you, that's exactly what I needed! I found an online table/calculator here. — kwami (talk) 07:54, 8 January 2014 (UTC)[reply]

optimum lengths for a gauge block set

Has there been any mathematical study of the "optimum" lengths for a gauge block set? What do you call such a set of lengths?

I'm looking for something analogous the way different ways of "optimizing" the marks on a ruler lead to different kinds of sparse ruler. And analogous to the way various ways of "optimizing" other sets of things leads to various sets of numbers called preferred number. --DavidCary (talk) 20:02, 8 January 2014 (UTC)[reply]

I don't know about gauge blocks, but at least under some kind of optimality criteria, optimizing coin denominations to make change making more efficient has been considered. Check out [1], which considers how to minimize the average number of coins needed to make change. If you wanted to instead minimize the number of uniquely sized gauge blocks needed to measure every integral length

1,\ldots ,N

, using a set of gauge blocks of lengths

1,2,4,8,\ldots

seems fairly minimal to me. --Mark viking (talk) 21:05, 8 January 2014 (UTC)[reply]

I was going to suggest the 1,2,4,8,... sequence myself. However, that does require a large number of different-sized blocks, and the math to figure out the total might be a bit much for some to do in their heads. A 1,10,100,... sequence would have fewer sizes of blocks, and figuring out the total in your head would be far easier, but you would need at least 9 of each size, so more blocks total, and you might have more cumulative error from using so many blocks. StuRat (talk) 21:17, 8 January 2014 (UTC)[reply]