Testwiki:Reference desk/Archives/Mathematics/2010 October 5

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October 5

Trying to find a recent question in the archives

I'm trying to follow a chain of links I explored in the not-too-distant past, but it's one of those things where I can't quite remember enough about it to get started. I think it started with a question asked here about a differential equation or something, and the questioner asked why the solution to the differential equation given in one of our articles contained the variable y as well as x. If I am correctly remembering where I read that, it should be here in the archives somewhere (probably in August or September), but I'm having trouble finding it. Does this ring a bell for anyone? —Bkell (talk) 02:59, 5 October 2010 (UTC)

Oh, I found it: Wikipedia:Reference desk/Archives/Mathematics/2010 September 9#Fitting a parabola and Euler's theorem. —Bkell (talk) 03:04, 5 October 2010 (UTC)

Probability: number of die throws necessary to reach a certain probability of covering all numbers

How do I calculate this:

I have a normal six-sided die. How many times do I need to throw the die to have at least at 50% probability of getting each number (1 through 6) at least once, in any order? How about if the die has 'n' sides and I want a probability of x%?

Thanks in advance219.164.239.187 (talk) 09:37, 5 October 2010 (UTC)

The probability that m independent throws of an n-sided die include all sides is $\frac{n!}{n^m}S(m,n)$, where S(m,n) is the Stirling number of the second kind.—Emil J. 11:54, 5 October 2010 (UTC)

Using the Inclusion–exclusion principle, if my calculations are correct, we can find that with m throws of an n-sided die, the probability of getting all outcomes is $\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k}){(1-\frac{k}{n})}^m$. Finding m given x requires solving a non-algebraic equation, but for the case $n=6,x=50$ you can numerically find $m\approx 12.811$ (so $m=12$ gives 43.78%, and $m=13$ gives 51.39%). -- Meni Rosenfeld (talk) 11:57, 5 October 2010 (UTC)

See also the coupon collector's problem. --Tardis (talk) 15:44, 5 October 2010 (UTC)

Thanks, you people are good! I've got it now.219.164.239.187 (talk) 00:44, 6 October 2010 (UTC)

Truncated lognormal

Hi. This is not homework. Say X is a lognormal variable (with $\mu$ and ${\sigma }^2$) and $Y=(X|X>A)$ where A is some positive number. I would like to derive experssions for $E[Y]$ and $Var[Y]$. I know this can be done using the Truncated normal distribution but I seem to be getting stuck. Can you help? Thanks. --Mudupie (talk) 11:26, 5 October 2010 (UTC)

Mathematica gives

$\underset{A}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{(\ln x-\mu )}^2}{2{\sigma }^2}}dx=\frac{1}{2}(1+\mathrm{erf}\left(\frac{\mu -\ln A}{\sqrt{2}\sigma }\right))$

$\underset{A}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{(\ln x-\mu )}^2}{2{\sigma }^2}}xdx=\frac{1}{2}{e}^{\mu +{\sigma }^2/2}(1+\mathrm{erf}\left(\frac{\mu +{\sigma }^2-\ln A}{\sqrt{2}\sigma }\right))$

$\underset{A}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{{(\ln x-\mu )}^2}{2{\sigma }^2}}{x}^{2}dx=\frac{1}{2}{e}^{2(\mu +{\sigma }^2)}(1+\mathrm{erf}\left(\frac{\mu +2{\sigma }^2-\ln A}{\sqrt{2}\sigma }\right))$

From which you should be able to find the mean and the variance. -- Meni Rosenfeld (talk) 12:37, 5 October 2010 (UTC)

Thank you very much for your help. --Mudupie (talk) 08:08, 7 October 2010 (UTC)

Probability

what is probablity ?make me understand how do question —Preceding unsigned comment added by 117.199.115.19 (talk) 13:22, 5 October 2010 (UTC)

See Probability. -- Meni Rosenfeld (talk) 13:56, 5 October 2010 (UTC)

If you check out the article that Meni linked to, you will see that there are two different ways of looking at probability: The frequentist approach and the Bayesian approach. From the way you formulate your question, I'm pretty sure it's the frequentist approach that you'll want to start with. In its easiest formulation, it goes like this: if you do an experiment, with (say) six different mutually exclusive and equally likely outcomes, the probability of each outcome is one sixth. This implies that if you repeat the experiment a large number of times (for instance observing the frequency of throws of a die that result in a six), the frequency will approach the probability. That is why it is called the frequentist approach. The sum of the probabilities of all possible outcomes is defined to be one. From this definition, and from a few other simple axioms, you can develop a set of rules for calculating the probability of composite events. You can use the fact that the frequency approaches the probability "in reverse", i.e. estimate the probability of an event from the observed frequency. --NorwegianBlue ^talk 21:15, 5 October 2010 (UTC)