Wikipedia:Reference desk/Archives/Mathematics/2010 March 28

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March 28

How much should my game cost?

Say I developed a games for the PC which I wishes to sell by mailorder.

Here are my values

FixedCost = $100000 (I send 100 000 dollars in a year to develop the game)

MarginalCost = $3 (it cost me $3 to stamp out a DVD-ROM and mail it )

DemandCurve or SalesVolume(price) =1000000*(1/(1+price)) (The amount of game units sold for each price in dollars)

When I tried to calculate the sales price for my game (to maximize my profits). My calculations kept on crashing saying I should sell my game at $999 999 dollars per game which is a ridiculous amount for sell a computer game to teenagers. What is wrong with my calculations?

122.107.207.98 (talk) 03:10, 28 March 2010 (UTC)

I get that there's no right answer, which indicates that your demand curve isn't realistic. We realize that (for x=price) total profit = income-expense = (x-3)*1000000*(1/(1+x))-100000. We want to maximize this number, so we take the derivative to find out how profit depends on price. We get dprofit/dx = 4000000/(1+x)^2 (sorry, I can't use the math type proficiently), which we see is always positive. That means that if you increase the price, you'll make more money, for all prices... not realistic. You can see that your demand curve isn't realistic by the limiting case: notice that as price increases to infinity, demand drops, not to zero (like in real life), but to one. Buddy431 (talk) 03:44, 28 March 2010 (UTC)

The demand does tend to 0 as $p\to \infty$. But yes, the demand curve is not realistic.

My bad. Buddy431 (talk) 21:27, 28 March 2010 (UTC)

Maybe there are consultants with experience on how the demand curves for this kind of game look like. Otherwise I suggest picking a reasonable number like $10 rather than making accurate calculations on made up data - Garbage In, Garbage Out.

BTW, have you considered some form of digital distribution? -- Meni Rosenfeld (talk) 08:14, 28 March 2010 (UTC)

To expand on "picking a reasonable number like $10", you can look at what competitors charge for comparable products and assume that your cost structure will be similar to theirs. That may sound like a lot of assumptions, but there are also a lot of assumptions which go into the demand curve and even the fixed and variable cost curves. So, the formal method really isn't any more accurate than "just guessing". StuRat (talk) 14:02, 28 March 2010 (UTC)

Tricky trig integral

Is there a way to evaulate $\int \sin ^{4}x\cos ^{6}x\;dx$ without the use of general $\int \sin ^{m}x\cos ^{n}x\;dx$ reduction formulae? I've been able to come up with concrete methods of how to evaluate all integrals of the form $\int \sin ^{m}x\cos ^{n}x\;dx$ by using the Pythagorean and double angle identities alone except for the ones where $m$ and $n$ are nonequal even powers both greater than or equal to four. Could someone help me evaluate the integral above and ones like it using only trig identities and u-substitution? Thank you! Yakeyglee (talk) 04:35, 28 March 2010 (UTC)

$\sin ^{4}x\cos ^{6}x=\left({\frac {\sin(2x)}{2}}\right)^{4}\left({\frac {1+\cos(2x)}{2}}\right)^{2}$

and then use one of the other methods which you have got. (Igny (talk) 04:59, 28 March 2010 (UTC))

I don't see how that helps to evaluate it if it's an integral. :-/ Yakeyglee (talk) 05:11, 28 March 2010 (UTC)

$\int \sin ^{4}x\cos ^{6}x\,dx=\int \left({\frac {\sin(2x)}{2}}\right)^{4}\left({\frac {1+\cos(2x)}{2}}\right)^{2}\,dx$

Multiply out, use substitution $u=2x$ and voila, you reduced the power, and you can use some of the methods which you said you already have. (Igny (talk) 05:15, 28 March 2010 (UTC))

How to tell if a time series is nonlinear or not?

I have some time-series I am interested in. How can I tell if they are non-linear or not? If they are not non-linear, then it means that I will not have to buy and read a thick expensive text-book about non-linear time series. Thanks 78.144.250.185 (talk) 15:47, 28 March 2010 (UTC)

To be honest, this is the very first time I heard about this topic. The third section of this paper may be a good place to start reading about NL testing. Pallida  Mors 17:10, 29 March 2010 (UTC)

Depends on what kind of non-linearity. If the model is non-linear in parameters, you'll need to use more complicated techniques to analyze the data. If the model is non-linear in regressors, you can simply try some non-linear forms of the regressors alongside the linear forms and see which show up as significant. Also, check the Box-Cox power transform. Wikiant (talk) 18:01, 29 March 2010 (UTC)

If I used nonlinear methods on a linear time series, would I still get valid results? Is it true that everything must be either linear OR nonlinear, or can common time series have elements of both? Thanks 78.146.84.14 (talk) 21:39, 29 March 2010 (UTC)

Assuming that your non-linear specification is a superset of the linear specification (for example, the true relationship is Y = a + b X + u, and you estimate the relationship Y = a + b X + c X^2 + u), then your estimates will be unbiased and consistent though inefficient. Wikiant (talk) 18:55, 1 April 2010 (UTC)

Probability

OK so this is homework but I have done my best to figure it out and still can't solve this.

A and B are independent. P(A union B) = 5/8. P(A intersect B') = 7/24.

First question was to find P(B) which I found by P(A union B) - P(A intersect B') = 1/3.

Second question is to find P(A intersect B). That's the one I'm struggling with.

Third question is to find P(A) which I know I can solve once I solve the second question. Since they are independent then P(A intersect B) = P(A) X P(B) so to find P(B), divide answer of second question by 1/3. I also could use the formula P(A) = P(A intersect B) + P(A union B) - P(B).

The given answer is 7/48 but I need to understand how to work things out! Please help quickly, thanks! —Preceding unsigned comment added by 59.189.218.247 (talk) 16:22, 28 March 2010 (UTC)

have you made a typo, or are you having one of those (all too frequent with me) 'duh' moments? P(A intersect B) is a given in the problem, it equals 7/24. --Ludwigs2 16:36, 28 March 2010 (UTC)

No, what's given is P(A∩B′), where B′ means the complement of B. —Bkell (talk) 16:41, 28 March 2010 (UTC)

you're right. sorry, misread it. --Ludwigs2 16:57, 28 March 2010 (UTC)

(ec) You have all the pieces you need; you just need to put them together. One way to find P(A) is to use both of the two formulas you've given, P(A)=P(A∩B)+P(A∪B)−P(B) and P(A∩B)=P(A)×P(B), and put in everything you know. You don't know P(A∩B) yet, but you can write it as P(A)×P(B). This will give you an equation which you can solve for P(A). Once you have that, it's easy to get P(A∩B). Another way to go about it is to reason like this: Since P(B)=1/3, and A and B are independent, then P(A∩B) should be 1/3 of P(A). That means that P(A∩B′) should be 2/3 of P(A). You know P(A∩B′), so you're set. —Bkell (talk) 16:40, 28 March 2010 (UTC)

Maybe I'm not seeing something, but I reckon there isn't enough information here to solve the problem. Imagine 4 area which sum to 1: AB=1/24, A'B=7/24, AB'=7/24, A'B'=9/24. Now imagine AB=7/24, A'B=1/24, AB'=7/24, A'B'=9/24. Both (and an inifinte number of other possibilities) meet the requirements with no unique answer. -- SGBailey (talk) 13:24, 30 March 2010 (UTC)

A and B are independent, so P(A∩B) must equal P(A)P(B). Neither of your examples satisfy this condition. —Bkell (talk) 14:21, 30 March 2010 (UTC)

Ah yes. So instead we have AB=3.5/24, A'B=4.5/24, AB'=7/24, A'B'=9/24. Which does satisfy the requirements. -- SGBailey (talk) 22:40, 30 March 2010 (UTC)