Wikipedia:Reference desk/Archives/Mathematics/2009 June 21

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June 21

Finding the median of a continuous random variable

I am told that I need to solve $\int _{m}^{\infty }f(x)\,dx=0.5$, and in the textbook example they get a polynomial with only m⁴, m² and m⁰ terms in it, which lets them solve a quadratic in m², take a square root and compare the results to the range which you are given at the start to find the right one, but I keep getting equations like $\int _{m}^{3}{\frac {2}{9}}x\left(3-x\right)\,dx=0.5$, where the p.d.f f(x) of the continuous random variable X was the above for $0\leq x\leq 3$, which eventually gets me to ${\frac {2}{27}}m^{3}-{\frac {1}{3}}m^{2}+0.5=0$, but I don't know how to solve this. In another example, I had the p.d.f. f(y) of the continuous random variable Y being $12y^{2}\left(1-y\right)$ for $0\leq y\leq 1$, which got me to $\int _{m}^{1}12y^{2}\left(1-y\right)\,dy=0.5\Rightarrow 3m^{4}-4m^{3}+0.5=0$, but, again, I don't know what to do with this. It Is Me Here ^{t / c} 08:58, 21 June 2009 (UTC)

Well, $\int _{0}^{3}{\frac {2}{9}}x\left(3-x\right)\,dx=1$, and the symmetry of $x\left(3-x\right)$ suggests that

$\int _{0}^{3/2}{\frac {2}{9}}x\left(3-x\right)\,dx=\int _{3/2}^{3}{\frac {2}{9}}x\left(3-x\right)\,dx=0.5$

so try m = 3/2 in your first problem. Your second problem is more difficult, as $y^{2}\left(1-y\right)$ doesn't have an obvious symmetry to exploit. Gandalf61 (talk) 12:21, 21 June 2009 (UTC)

What's worse, Mathematica seems unable to find a representation of the solution simpler than direct substitution in the quartic formula. Is there any chance they want you to find a numerical solution (which is 0.614272...)?

By the way, the first problem can also be solved by the rational root theorem. -- Meni Rosenfeld (talk) 13:21, 21 June 2009 (UTC)

See root-finding algorithm for methods to solve equations numerically. Or draw the conclusion that the median is not that interesting anyway and use the mean value instead! Your examples are beta distributions. By the way, 0.614272 is an approximate solution to $3m^{4}-4m^{3}+0.5=0$. The J program p. 1 0 0 _8 6 does the trick. Bo Jacoby (talk) 13:42, 21 June 2009 (UTC).

OK, the actual question states: "The continuous random variable Y has p.d.f. f(y) defined by f(y) = 12y²(1 - y) for 0 ≤ y ≤ 1; f(y) = 0 otherwise. Show that, to 2 decimal places, the median value of Y is 0.61." It Is Me Here ^{t / c} 15:09, 21 June 2009 (UTC)

That's easy then. You know that $F(y)$ (the cumulative distribution function) is continuous and increasing, so you only need to show that $F(0.605)<0.5$ and $F(0.615)>0.5$. The intermediate value theorem will then say that there is a root $0.605<y<0.615$, which is thus equal to 0.61 to two decimal places. -- Meni Rosenfeld (talk) 18:47, 21 June 2009 (UTC)

Induction

Not a massive fan of induction and so I would like to check my solution to a problem with you guys.

"Let $S_{n}(x)=e^{x^{3}}{\frac {d^{n}}{dx^{n}}}(e^{-x^{3}})$. Prove by induction on n that $S_{n}(x)$ is a polynomial."

Assume that $S_{n}(x)$ is a polynomial for n=k.

$e^{x^{3}}{\frac {d^{k}}{dx^{k}}}(e^{-x^{3}})=f(x)$

So ${\frac {d^{k}}{dx^{k}}}(e^{-x^{3}})=e^{-x^{3}}f(x)$

Differentiate the above wrt x.

${\frac {d^{k+1}}{dx^{k+1}}}(e^{-x^{3}})=e^{-x^{3}}f'(x)-3x^{2}e^{-x^{3}}f(x)$

Factorise

${\frac {d^{k+1}}{dx^{k+1}}}(e^{-x^{3}})=e^{-x^{3}}(f'(x)-3x^{2}f(x))$

$f'(x)-3x^{2}f(x)$ is still a polynomial for any f(x), so let us call it g(x).

So we have

${\frac {d^{k+1}}{dx^{k+1}}}(e^{-x^{3}})=e^{-x^{3}}g(x)$

So if the result is true for k, it is also true for k+1. Now let n=0 (the question doesn't actually state which set of numbers 'n' belongs to but I assume it's the non-negative integers), which gives us the case where $S_{0}(x)=1$. So by induction, $S_{n}(x)$ is a polynomial for all integers n, n≥0.

Is that airtight? Thanks. asyndeton talk 14:15, 21 June 2009 (UTC)

Yes. —JAO • T • C 14:42, 21 June 2009 (UTC)

It looks fine to me, although I would have started with n=1. While defining the zeroth derivative to be the identity makes sense, I would normally think of n be positive in the context of n-th derivatives. --Tango (talk) 14:56, 21 June 2009 (UTC)

well, I think yours is mainly a psycological limit. But if you work on it a bit, you too will be able to start with n=0  ;-) --pma (talk) 19:11, 21 June 2009 (UTC)

It's obvious that is does work with n=0, I just don't normally think of the identify being a derivative. It's a matter of definition, rather than any mathematical, of course. --Tango (talk) 00:05, 22 June 2009 (UTC)

The definition A⁰ = 1 apply very generally. See exponentiation. And so $\scriptstyle {\frac {d^{0}}{dx^{0}}}=1$ is what you should normally think. What happens if you don't differentiate? The same thing as if you multiply by one. Nothing happens! Bo Jacoby (talk) 05:08, 22 June 2009 (UTC).

When you say $\scriptstyle {\frac {d^{0}}{dx^{0}}}=1$, do you mean this is true in general no matter what it is you're differentiating or is it just true for what I gave above? asyndeton talk 07:44, 22 June 2009 (UTC)

Yes, in general, for a linear operator A, the useful notation is A⁰=I (the identity operator, that we think to as the multiplication by 1 and write indeed 1) and A^k=A...A k-times for all positive integers k. --pma (talk) 10:06, 22 June 2009 (UTC)

In case there is any confusion, note that it is $\scriptstyle {\frac {d^{0}}{dx^{0}}}=1$, not $\scriptstyle {\frac {d^{0}}{dx^{0}}}f(x)=1$. It's the identity operator, not a constant operator. --Tango (talk) 15:09, 22 June 2009 (UTC)

I don't see the difference between the two. asyndeton talk 16:54, 22 June 2009 (UTC)

The second one is $\scriptstyle {\frac {d^{0}}{dx^{0}}}f(x)=1\cdot f(x)=f(x)$. Bo Jacoby (talk) 17:05, 22 June 2009 (UTC).

The first is an operator, the second is an operator acting on a function. When we say an operator is "1" we mean it is the identity - it maps everything to itself. That's very different to saying the result of the operator acting on a function is always equal to the number "1". --Tango (talk) 17:08, 22 June 2009 (UTC)

N-deck hearts

Do the rules of double-deck cancellation hearts generalize well to hearts games with N decks and 4N-2 to 4N+2 players? NeonMerlin 23:46, 21 June 2009 (UTC)

Well my first thought is that there are two possible ways in which the game could be generalised, either we stick with the rule that if a second identical card is played then the two cards cancel, which leads to a game where if odd numbers of an identical card are played the last player to play that card (assuming it is the strongest one, otherwise it makes no difference) takes the trick whereas if even numbers are played the cards are ignored, this seems slightly artificial to me and as if it would lead to a confusing game.

Alternatively the other way in which the game could be generalized is to only apply the cancellation rule when all N of an identical card are played in one trick, otherwise if say K (<N) identical cards were played in one trick then the last player to play the identical card would count as having played it and the other players who played it would be ignored. This to some extent removes the excitement that double-deck cancelation hearts has, since especially when N is large the chance of a cancellation occuring are very slim.

Both possibilities have their pros and cons, when N is small (say 3 or 4) I think both variations would make interesting playing. —Preceding unsigned comment added by 86.129.82.211 (talk) 00:52, 22 June 2009 (UTC)