This here being a homework educational value thread probably one must never miss the chance to expand on these things and we can all learn more that way;

For example try also these links with detailed analysis and generalizations

http://www.coe.utah.edu/~cs5960-02/l...s/Lecture2.pdf

http://stats.stackexchange.com/quest...ndom-variables

Look also in the related product entry in the general link

https://en.wikipedia.org/wiki/Variance



https://www.jstor.org/stable/2281592

and


if independent.

notice



https://en.wikipedia.org/wiki/Covariance

This is worthy too

http://www.stat.yale.edu/~pollard/Co...7/Variance.pdf



Additional value

http://www.math.uah.edu/stat/dist/Joint.html

https://en.wikipedia.org/wiki/Product_distribution

(forgot password to main account, sorry if this is against the rules)

I have a stats problem, happy with any direction people can point me in. For context I've done intro mathematical stats so I know how to do easier functions of random variable questions (CDF method, transformation method, mgf method) but I'm lost here.

Question:

u and v are i.i.d sampled from the uniform distribution [0,1]

if s^2 = (2u-1)^2 + (2v-1)^2 < 1, then x is sampled from the standard normal distribution. Otherwise, x = (2u-1) * sqrt( -2log(s)/s)

My first thought was to try and use the cdf method and then glue them together in a pdf, I'm not getting very far. The other problem I have is that it seems to me that the conditions imposed on s (if the square is less than 1) imply that the 'otherwise' section would result in a negative log.

Oh should have added: how is x distributed?

Anyway, remembered pw to this account

Not homework but something I wanted to figure out & I'm a complete drooler so maybe you guys could help me out. I have five people processing returns from several (call it 100) different companies every day. The companies take a different amount of time to process so 15 units of company A might take the same amount of time to process as 40 units of company B. Having records of how much of each company everyone did each day how do I tease out who's outperforming & who is just consistently grabbing the companies that get be processed fast to juice their total numbers?

It seems that you should be able to get an average processing time by company from the data. And then you can benchmark against that.

Factorisation:

I am having trouble breaking down this expression in the way that is wanted. I am asked to factorise an expression using the difference of two squares.

(squared = (s))

3x(s) - 27y(s) = ?

The answer is 3(x-3y)(x+3y)

I have tried a lot of different ways of re-expressing it but I can not get the maths to compute to that.

Let's pull out a 3 first to get 3x^2-27y^2 = 3(x^2-9y^2)

Now, you hopefully know that (a-b)(a+b) = a^2-b^2. If you set a^2=x^2 and b^2=9y^2, I think that you can find some pairs of square roots to find what you need.

I think this is an ANOVA (analysis of variance) question. From what I can recall from my stat lecture 10 years ago.

I'm taking a maths course this summer, and apparently my geometry is a lot worse than I thought. Can someone help me with this?

Draw a parallellogram ABCD. From A, draw the line AN perpendicular to CD and the line AM perpendicular to BC (N lies on the line CD, and M lies on the line BC). Show that MAN ~ ABC (not quite sure what this translates to, but that the triangles MAN and ABC are similar I think?)



Any help would be greatly appreciated, I'm feeling like an idiot here.

Call ABC angle x.

ABC is the same angle as MAN. Triangles ABM and AND are similar right triangles (they have same angles NDA is ABM ie both 180-x.

Then DAB is 180-x, ABM is 180-x so BAM is x-90 and NAD also x-90.

So MAN angle is 180-x+x-90+x-90=x ie the same as ABC angle.

From right triangles above mentioned being similar you get AN/AD = AM/AB =AN/BC.

So you have two triangles that have one common angle and their sides that form that angle are proportional to the same corresponding sides of the other triangle (same ratio). So they have to be similar.

To show that, imagine drawing the 2 triangles (MAN and ABC) with common angle ABC and a line parallel to NM from a point that is AB/AM away from A (on AM extended direction line) as AM is from A (ie the point B lol). Then that line on the other side crosses to point C basically creating the identical to ABC triangle (imagine it crosses to C' and prove it has to be C because of similar triangle properties as the just created triangle is similar to MAN and you can prove AC=AC').

Thanks a lot masque, I really appreciate the help. Hopefully the rest will come easier to me, otherwise I might be back

Trig problem, probably something silly I’m not getting that I just need spelled out for me:

“Find tan(17pi/12) using the addition identity of two special angles”

The correct answer is (sqrt(3) + 1) / (sqrt(3) – 1) but I get something slightly different:

When I do the problem I get:

Tan(17pi/12) = tan(9pi + 8pi/12) -> 3pi/4 + 2pi/3

Sin(3pi/4 + 2pi/3) = sin(3pi/4)cos(2pi/3) + cos(3pi/4)sin(2pi/3)

Cos(3pi/4 + 2pi/3) = cos(3pi/4)cos(2pi/3) – sin(3pi/4)sin(2pi/3)

... (sin(X + Y) * 1/(cosXcosY)) / (cos(X + Y) * 1/(cosXcosY)) …

= tan(3pi/4) + tan(2pi/3) / 1 – tan(3pi/4)tan(2pi/3)

= (-1 - sqrt(3)) / 1 + sqrt(3))

So for some reason I am getting the negative of the correct numerator and the conjugate of the correct denominator in my answer...

That should be ;

tan(3pi/4) + tan(2pi/3) / 1 – tan(3pi/4)tan(2pi/3)

= (-1 - sqrt(3)) / (1 - sqrt(3)) (- - - three times = -)

which is (1+√3)/(√3-1) as wanted.

Somehow I managed to stare at this answer for like 15 minutes and not realize that. Thanks, knew it was something dumb.

Statisticians:

I've been presented the following problem from work (so I may get approval to compensate if someone can produce a comprehensive report, PM me if you're interested and we can discuss. I'd like to give someone that opportunity before I produce the whole dataset):

We have a powerball style game where a machine contains 40 balls and, 1 by 1 and without replacement, it selects 6 balls. One 6-of-40 selection constitutes a single trial.

The balls were weighed, and the goal is to determine if the variations in weights produce any "unfairness" in the frequency of occurrences. Max weight 4.668g / Min weight 4.534g / population variance 0.0011209g

Would anyone like to take a look at the data and determine if any balls occur too frequently, not enough, too frequently in combination with (an)other ball(s)? Is coupon collector relevant here? What other tests might be predictive of unfairness?

I have data for 301 trials (1806 balls drawn) which is obviously a tiny sample. Is it too small to draw meaningful conclusions?

I appreciate your time.

If this post is not OK because I'm not trying to give a reasonable effort at solving, my apologies. Can someone point me in the right direction?

Way too small of a sample since each ball is, on average, going to show up only 45 times. You'll have a large variance for each of the balls appearing.

Thank you

Do you actually know the exact weight of each ball and their occurrence? The sample is too small at first glance especially if other factors change from run to run. On the other hand if you have persistently a bias towards a certain weight of balls you can for example separate the balls to the 50% top heavier group and 50% lower and then it is quite possible you will see a bias because in 1806 calls it ought to be 50-50 and if you see some significant variation for one group from that then you are in business. You can do the same with top 25% etc. I think its possible to see things if you study the group behavior. Nothing conclusive in a big way but possibly validating a claim to further study with confidence.

Try for example top 50% top 40% top 30% and if always the heavier (or whatever ) comes above expected it starts looking interesting.

Yes I have the weight of each ball and the balls drawn in each of the 300 occurrences or trials.

I like your methodology I think. I'm hopelessly unqualified to perform the tests myself, though.

You also have the sequence the balls come out per event. That too may matter a lot.

The optimist in me and the AI new age super intelligence futurist in me lol suggest you can find out a ton of info from all kinds of angles together imagined.

I do not however have a reason to think the small variation in mass (at most 3%) will lead to such probability biases that are significant enough to prove visible in small samples.

mdZ, I sent you a couple PMs. My feelings won't be hurt if you ignore.

Do you even know if the weight affects whether or not balls are selected? Who's to say if heavier balls are more or less likely to be picked, or if it really doesn't matter.

Obviously we don't know; that's the point of the exercise. We don't quite have hundreds of millions of dollars riding on this project like a traditional state lottery might, but our exposure is somewhat significant.

Do you have any other ideas about what might affect frequency besides weight? What are you suggesting in your post?

I would record the weight of each ball and make a chart of frequency vs weight and see if it's obviously trending up or down.

I have the weight of each ball and a chart of actual occurrences