Wikipedia:Reference desk/Archives/Mathematics/2017 February 13

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February 13

Standard deviation of a product

If the stochastic variables $X\pm Y$ and $Z\pm W$ are independent, then the sum is

$(X\pm Y)+(Z\pm W)=X+Z\pm {\sqrt {Y^{2}+W^{2}}}$.

This is well known. I did not find the corresponding formula for the product, in wikipedia or elsewhere, so I had to derive it myself:

$(X\pm Y)(Z\pm W)=XZ\pm {\sqrt {X^{2}W^{2}+Y^{2}Z^{2}+Y^{2}W^{2}}}$.

Did you know this formula? Please provide a reference. Bo Jacoby (talk) 01:54, 13 February 2017 (UTC).

Could you clarify the question? The section header mentions standard deviation, but standard deviation appears nowhere in your question. Also, your first equation seems to state that $\pm Y\pm W=\pm {\sqrt {Y^{2}+W^{2}}},$ which I don't understand. Are some the symbols supposed to be standard deviations? Loraof (talk) 16:13, 13 February 2017 (UTC)

I don't know if it's relevant to you, but we have Variance#Product of independent variables. Loraof (talk) 16:13, 13 February 2017 (UTC)

Thank you! The link you gave me is exactly what I was looking for. The notation $X\pm Y$ indicates a variable with mean value $X$ and standard deviation $Y$. Bo Jacoby (talk) 17:30, 13 February 2017 (UTC).

The cases $X=Z=0$ give the nice formulas

$(\pm Y)+(\pm W)=\pm {\sqrt {Y^{2}+W^{2}}}$.

$(\pm Y)(\pm W)=\pm {\sqrt {Y^{2}W^{2}}}$.

Bo Jacoby (talk) 07:03, 14 February 2017 (UTC).

And the last one simplifies to the (surprising to me) "the standard deviation of the product of independent zero-mean variables equals the product of the standard deviations." Loraof (talk) 21:07, 14 February 2017 (UTC)

Yes, that was surprising to me too! And the computation of

$(X\pm Y)(Z\pm W)$

has the beautiful intermediate result

$XZ\pm XW\pm YZ\pm YW$

which by the sum rule

$\pm A\pm B=\pm {\sqrt {A^{2}+B^{2}}}$.

evaluates to

$XZ\pm {\sqrt {(XW)^{2}+(YZ)^{2}+(YW)^{2}}}$

Bo Jacoby (talk) 01:19, 15 February 2017 (UTC).

Is a Jupiter-, then Saturn-grazing approach to Earth possible for Interstellar ship under constant deceleration?

Imagine a ship entering the solar system under a constant deceleration, with the eventual goal of entering Earth orbit? Ignore the gravitational (possible slingshot) effects of Jupiter and Saturn, but not of the Sun, or Earth when relevant--the craft is under power and the aaliens want this path and this deceleration and will spend energy to compensate.

(1) What would be the overall shape of such an approach, ignoring final orbit? Parabolic? Spiral? Other?

(2) Would an approach under constant deceleration (see my speculation below) be possible that would bring the incoming craft past Jupiter first, then Saturn, ending in earth orbit?

My first feeling (visual thinking, I don't know the math) is that it would be impossible, since Saturn is about twice as far out as Jupiter. Any approach that had already passed Jupiter would have to be headed inward.

Then I wondered, assuming the planets are properly aligned, if a parabolic route falling into the solar system along the ecliptic from 12 o'clock might pass Jupiter when it is at 9 o'clock, head out to Saturn at 6 o'clock, (near the vertex of the parabola), then fall back into the solar system approaching Earth at around 3 o'clock (although where, exactly, doesn't matter)? (My guess here was that since the derivative of a parabola is a constant, a constant deceleration might allow a parabolic approach that, with luck, would pass Jupiter at speed while slowing down at Saturn and falling in towards the Earth.)

Thanks. μηδείς (talk) 01:55, 13 February 2017 (UTC)

If there were no retro-engines firing, the path of an object approaching the Earth while deflected by the gravity of the Sun should be a conic section. Specifically it would be a parabola since it wouldn't fall into orbit that way. But the ship's engines could change that into an elliptical orbit, and even a circular orbit eventually. Exactly how much and when the engines fire might make the trajectory discontinuous in curvature, though, not the smooth curve an object without engines would have. StuRat (talk) 05:03, 13 February 2017 (UTC)

Basically what I want is an alien craft that can come close enough to do a survey of Jupiter, and then Saturn, in that order, then orbit Earth with minimal use of thrusters at each point to maintain a constant deceleration that matches the crew's normal gravity. I know a constant deceleration will achieve this. I know that there may need to be adjustments to overcome the local planetary slingshot effects; I want to minimize those. Will a basically parabolic orbit allow this given my prior criteria?

Can someone plot this type of system entry, showing a mostly parabolic path will work, assuming the planets are properly aligned at the point of insertion?

I know that the Math Desk people are smaarter than I am, and hope a graphic can accomplish a proof. μηδείς (talk) 05:58, 13 February 2017 (UTC)

That's rather complex, because not only would you need to know the positions of the planets when you start, but each is moving as your ship moves through the solar system. You may need somebody at NASA to help you. StuRat (talk) 06:27, 13 February 2017 (UTC)

I have already specified this ("assuming the planets are properly aligned"); I want an arbitrary solution, not one based on any specific date. The only constraints are that the craft remain under constant deceleration, and that it pass Jupiter before it passes Saturn, and then approaches the earth. All other factors other than the Sun's gravity can be chosen for convenience, and the approaches to Jupiter and Saturn should be assumed to be distant enough so as not to factor largely in the calculations or the craft's minor course corrections.

I already know all about conic sections and derivatives, I took AP Calculus and tested out of 12 credits of math toward my biology major as an undergrad. What I am not sure about is any other issues of calculating powered trajectories and orbital mechanics; my thought on this has been visual, and I was hoping for a rigorous mathematical treatment, or at least an authoritative statement with some further reading that backs it up. Thanks. μηδείς (talk) 19:12, 13 February 2017 (UTC)

I don't believe constant, uniform deceleration would be possible. First, the direction of the deceleration vector would need to change, but also the magnitude would need to change as it neared Jupiter and Saturn, to get the proper deflection, to hit the next target trajectory. StuRat (talk) 18:50, 14 February 2017 (UTC)

I have already said that minor course corrections would be possible and that the approaches to the gas giants should be taken as not so close as to require hard corrections. The idea is having a ship maintain a normal "artificial" gravity, and if it has to change vector that is not a concern. Basically, you are bringing up objections that were considered in my first post. What I want to know is, given arbitrarily chosen locations in their orbits and ignoring or correcting for minor interactions, could an interstellar ship approach Jupiter first, then Satur, and then Earth in a trajectory that approaches a conic section without wild accelerations or so forth. I suspect a close-to parabolic approach to the Earth might do this, if Saturn and Jupiter were in the right locations. μηδείς (talk) 00:25, 15 February 2017 (UTC)

Even a mere 1m/s/s deceleration would bleed almost 100km/s a day. A direct beeline from Saturn to Earth at their closest approach (~8 AU) would still take 18 days and average c. 777.7 kilometers per second and exceed 100 times Saturn escape velocity at the start. Gravity would barely touch them for much of the trip (1 gravitational constant takes years just to turn Saturn) and they'd move in an almost straight line till they slow down enough unless they are pointing their retrorocket at different parts of the general direction of travel to steer. If this trip is possible then they'd make a path which I don't have the math skills to calculate or imagine. Sagittarian Milky Way (talk) 01:25, 15 February 2017 (UTC)

Sizes of polygonal lakes

Lake Pathological

Median line

At Talk:List of lakes in the Lake District there was discussion about the sizes of lakes in terms of "length" and "fetch". My dictionary defines fetch as "the distance travelled by a wind or wave without obstruction". This set me thinking about things geometrically based on a polygonal lake (blue). I think "fetch" is the longest straight line that can be drawn within the lake (red) and I was a bit surprised to find the ends are not necessarily at vertices ("e" is not a vertex). For length I imagine a fish swimming the shortest distance between the ends of the lake. The ends are those two points (I guess they will be vertices) between which the fish's swim is the greatest distance – AB (green) in this case. For a crow, A and B are close together and C and D are the furthest points apart. Nonetheless I feel that A and B are the "ends" of the lake. Another measure of length would be to consider a median line going between the ends (dashed black). By median I mean a line equidistant between the opposite shores. I had difficulty deciding what line to take, particularly round the north of the lake. I wonder whether median is well-defined for an arbitrary polygon – perhaps "opposite" is not well-defined to start with. It seems likely to me that it might be possible for a lake to have different "ends" depending on whether the swimming or median length was chosen. Does anyone have any observations on any of this? Thincat (talk) 11:22, 13 February 2017 (UTC)

The median path could be defined as that which maximized the distance to the nearest shore for each point on the path. Consider all circles that touch both shores (either side of the path), and that nowhere go onto land; then the median path would be locus of the centres of this set of circles. Where the circles touch either a straight segment of shore on both sides, or a vertex promontory on both sides, then the this path would be a straight line segment. Where the circles touch a straight shore on one side and a vertex promontory on the other (as at the north end of Lake Pathological), then the path would be a segment of a parabola.--catslash (talk) 14:51, 13 February 2017 (UTC)

Note that real lakes do not have vertices, or, equivalently, have an infinite amount, depending on how you think of it. It's the reverse situation of the coastline paradox. StuRat (talk) 15:13, 13 February 2017 (UTC)

See also Pick's theorem. — 79.115.132.168 (talk) 17:23, 13 February 2017 (UTC)

Thank you for these responses. It is a neat suggestion to define a median line by considering a variable circle sliding along the lake leading to a sequence of straight and parabolic segments. I think that is a surprisingly elegant result which I find very pleasing. Thank you. Thincat (talk) 09:27, 15 February 2017 (UTC)

Just for my own interest I have added a diagram (above) showing the median line (solid black). The centres of the fitted circles are simply joined with straight lines. I have kept the dashed line I had previously drawn by eye. Thincat (talk) 11:15, 15 February 2017 (UTC)

It is interesting to see this idea realized. If this is an unambiguous definition of median path, then there must be a definite answer to the question of whether the swimming length and the median length can have different ends. I can't think of an example for which the ends are different.

Regarding the longest fetch, I can think of examples for which neither end is on a vertex, but I reckon the line must touch at least one vertex, and suspect it must touch at least two. --catslash (talk) 22:42, 15 February 2017 (UTC)

I have just redrawn the median path diagram more accurately and with minimal circles – only where each straight segment ends and where each parabolic rotation around a concave vertex against a single straight line ends. I had to approximate the (short) parabolic segments with Bezier curves (Inkscape). I think the path is unambiguous in that there was never a dilemma as to where to place a circle and with what radius. Whether another definition of median could be sensibly used, I don't know, but your definition seems pretty much ideal to me. Thincat (talk) 23:48, 15 February 2017 (UTC)

Belle Isle in the middle of Windermere

.

There is one possible source of ambiguity - branches. In the example here, there's no way to know a priori which of the two inlets to go down. You'd have to do both, and then find which is longest. This would get especially complicated where islands are involved, and it might give some weird results. To stick with the Lake District, in the middle of the lake you have Belle Isle (Windermere). What's the best way to integrate that into this scheme? I'm not sure there is one. If you ignore the island, the "median route" through Windermere is one that cuts straight through it. This would be impossible to do in a boat, so it seems like a bad way to define the lake's length. So you're faced with a choice of going left or right. I suspect left would be shorter, since to go right involves turning through the bay at Bowness. What seems more natural here - to define the fetch by the length of the most direct route, or the most indirect one? If you're going for the longest possible median path, there are even convoluted routes where you go through Bowness Bay, cut around the headland of Belle Isle, go past Thompson Holme, weave around the Lilies of the Valley, and then keep going north up the other side of Thompson Holme. This is a longer median route, but it also doesn't seem like the right way to define the lake's length. Smurrayinchester 10:00, 16 February 2017 (UTC)

I have been browsing around but now I have now reached the limit of my mathematics. What I think is unambiguous is the longest path along the medial axis of a simple polygon, Chin, Francis (9 June 2005). "Finding the medial axis of a simple polygon in linear time". Algorithms and Computations, Lecture Notes in Computer Science. 1004: 382–391. doi:10.1007/BFb0015444. I agree it is necessary to investigate multiple pairs of end vertices. Yes, there are real-world ambiguities with islands, promontories, piers, etc. Suppose I announce the length of a lake and give the corresponding path which crosses an island. If you are a yachtsman you will need to take a longer route whereas if you are a geographer you may not care because the presence of islands arguably does not affect the length of a lake. Up until now, as well as never having come across "medial axis", I had not come across Pick's theorem (mentioned above) either. For such a simple, useful theorem (well, useful for measuring the areas of lakes with digital mapping) it surprises me it was not discovered before 1899. Thincat (talk) 14:10, 16 February 2017 (UTC)

Probability homework question

My daughter is taking a probability course and has a homework problem about the expected number of cards picked from a deck (without replacement) until a queen is drawn. She knows that the equation for R(N,k) near the top of the green area of this link (scroll down a little) is the correct formula, but doesn't see how it comes about. Can someone briefly explain why that is it? Bubba73 ^{You talkin' to me?} 16:55, 13 February 2017 (UTC)

I find it helpful to toss aside the notation and explain it in concrete terms. So, if we have 52 cards, and 4 are queens, then the chances of getting a queen on the first draw are 4/52. On the second draw, if we didn't find a queen in the first draw, the chances are 4/51. However, we need to account for that condition that it wasn't drawn the first time. So, since the chances of getting to a draw are the chances we didn't find the queen by the end of the previous draw, or 48/52 or 0.923, that means (0.923)(4/51) are the chances we would get to and draw the queen on the second draw. Let's chart this pattern:

DRAW   CHANCE OF           CHANCE OF       TOTAL CHANCE OF        CUMULATIVE CHANCE  
 #     GETTING HERE        DRAWING QUEEN   QUEEN THIS DRAW        WE DREW A QUEEN
----   -----------------   -------------   --------------------   -----------------
 1     1                    4/52           1.000(4/52) =  0.077           0.077
 2     1 - 0.077 = 0.923    4/51           0.923(4/51) =  0.072           0.149
 3     1 - 0.149 = 0.851    4/50           0.851(4/50) =  0.068           0.217

Try to continue this chart and see when you get to a 50% cumulative probability (you might want to use more decimal places than I did, for accuracy). Does this help her to understand how it works ? StuRat (talk) 17:27, 13 February 2017 (UTC)

I'll send it to her. I was thinking along those lines, but the cumulative probability isn't going to hit 0.5 exactly. Bubba73 ^{You talkin' to me?} 17:35, 13 February 2017 (UTC)

Seeing how many tries till you get to 50% gets you the median, not the expected value. For a proof of the formula, I like the proof by induction given in the link after the green section, down at the bottom. Loraof (talk) 17:42, 13 February 2017 (UTC)

Thanks, we didn't look down that far. That derivation is easier to understand than the one with all of the combinations. Bubba73 ^{You talkin' to me?} 17:58, 13 February 2017 (UTC)

You can think along the following lines. Let N be the number of cards, q – the number of queens and k – the number of draws. What is the number of ways (n) that you can draw k cards from the total of N cards? This is

$n={\frac {N!}{(N-k)!k!}}$

What is the number of ways you can draw k cards from N and not draw a queen? Or putting it differently: what is the number of ways to draw k cards from (N-q) cards that are not queens? This is

$n^{*}={\frac {(N-q)!}{(N-k-q)!k!}}$

The probability that you draw at least one queen after k draws ($p_{k}$) is obviously:

$p_{k}=1-{\frac {n^{*}}{n}}=1-{\frac {(N-q)!(N-k)!}{N!(N-k-q)!}}$

Ruslik_Zero 18:38, 13 February 2017 (UTC)

Thanks. But that doesn't give you the expected number of draws until you get a queen, does it? Bubba73 ^{You talkin' to me?} 18:53, 13 February 2017 (UTC)

The expected number of draws is

${\bar {k}}=\sum \limits _{k=1}^{N-q+1}k\,p_{k}(1-p_{k-1})$

Ruslik_Zero 19:54, 13 February 2017 (UTC)

Thanks, I think it is clear now. Bubba73 ^{You talkin' to me?} 20:18, 13 February 2017 (UTC)

Well, she got it on her own after a false start of doing it as if the card was replaced and the deck shuffled before each draw. Bubba73 ^{You talkin' to me?} 00:29, 14 February 2017 (UTC)

Resolved