Eqn w/ unknown variables & probability: Pi(H-R) - pi(1-pi)(H-R)^2 = pi(L-R) - pi(1-pi

[imredp]

Eqn w/ unknown variables & probability: Pi(H-R) - pi(1-pi)(H-R)^2 = pi(L-R) - pi(1-pi

I have an equation where I cannot find the R. I use the word pi instead of the icon, which is a probability! So no use is made of the number of pi.

Pi(H-R) - pi(1-pi)(H-R)^2 = pi(L-R) - pi(1-pi)(L-R)^2

According to my professor, R = (H+L)/2 - (1/2)/(1-pi)

I tried to get this answer several times but I keep failing again and again. I eliminated pi and then (1-pi), but I keep ending up with terms like H-L instead of H+L etc etc.

I hope someone is able to help me because it will be important for my final exam of my master.

Thanks in advance.

 

[Dr.Peterson]

I have an equation where I cannot find the R. I use the word pi instead of the icon, which is a probability! So no use is made of the number of pi.

Pi(H-R) - pi(1-pi)(H-R)^2 = pi(L-R) - pi(1-pi)(L-R)^2

According to my professor, R = (H+L)/2 - (1/2)/(1-pi)

I tried to get this answer several times but I keep failing again and again. I eliminated pi and then (1-pi), but I keep ending up with terms like H-L instead of H+L etc etc.

Please show us at least the main steps of your work, so we can look for a specific error you are making. I get the professor's result; but I don't know what it would mean to "eliminate pi". I expanded everything (but keeping (1 - pi) as a unit), and solved.

By the way, in my own work, to avoid confusion, I just used "p" instead of "pi".

 

[imredp]

I eliminated p by dividing both sides with p. I tried keeping (1-p) as a unit and I made good progress. However I ended up with: ((H-L)/(1-p))/(2L-2H) + (L^2-H^2)/(2L-2H). The second part I could derive to (H+L)/2, but the first part I ended up with (H-L)/(1-p)*(2L-2H) as a/b/c equals a/b*c. Now, according to my professor's answer, I should just divide H-L by -2H+2L. However, should you not divide H by (-2H+2L) and not only by -2H?

Thank you

 

[Dr.Peterson]

I eliminated p by dividing both sides with p. I tried keeping (1-p) as a unit and I made good progress. However I ended up with: ((H-L)/(1-p))/(2L-2H) + (L^2-H^2)/(2L-2H). The second part I could derive to (H+L)/2, but the first part I ended up with (H-L)/(1-p)*(2L-2H) as a/b/c equals a/b*c. Now, according to my professor's answer, I should just divide H-L by -2H+2L. However, should you not divide H by (-2H+2L) and not only by -2H?

Ah, good -- you didn't eliminate all p's, just the common factor of all terms.

So from p(H-R) - p(1-p)(H-R)^2 = p(L-R) - p(1-p)(L-R)^2
you got (H-R) - (1-p)(H-R)^2 = (L-R) - (1-p)(L-R)^2
and expanded to H - R - (1-p)(H^2 - 2HR + R^2) = L - R - (1-p)(L^2 - 2LR + R^2)
and then H - R - (1-p)H^2 + 2(1-p)HR - (1-p)R^2 = L - R - (1-p)L^2 + 2(1-p)LR - (1-p)R^2
and then perhaps canceled like terms, H - (1-p)H^2 + 2(1-p)HR = L - (1-p)L^2 + 2(1-p)LR
and moved the R terms to the right and others to the left, H - (1-p)H^2 - L + (1-p)L^2 = 2(1-p)LR - 2(1-p)HR
and factored, (H - L) + (1-p)(L^2 - H^2) = (1-p)(2L - 2H)R

Up to here, I think the only thing I'd do differently is to keep the 2 outside, so the right side would be 2(1-p)(L - H)R. It's helpful to keep things in factored form as long as there's no need to do otherwise, and this will make it easier to see the step you seem to have missed.

Then, the way I would write it, you divided by 2(1-p)(L - H), and simplified a bit to

R = (H - L)/[2(1-p)(L - H)] + (1-p)(L + H)(L - h)/[2(1-p)(L - H)]

which simplifies to R = (H - L)/[2(1-p)(L - H)] + (L - H)/2 = -1/[2(1-p)] + (L - H)/2, because (H - L) = -(L - H).

Why do you think he divided by -2H at any point?

A couple points to add: first, it isn't properly true that a/b/c equals a/b*c; both are read left to right, so that latter has to be written as a/(b*c) when you write on one line. Second, I avoided that entirely by dividing by [2(1-p)(L - H)] all at once.

 

[imredp]

I indeed ended up with (H - L) + (1-p)(L^2 - H^2) = (1-p)(2L - 2H)R . Now when you propose to keep the 2 outside, it is indeed easier to divide. However, I see you ended up with -1/[2(1-p)] + (L - H)/2, where my professor's answer was: -1/[2(1-p)] + (L+H)/2. Is this because when we divide ((1-p)(L^2 - H^2)) by 2(1-p)(L - H), the minus cancel each other out? so that (L-H)*(L+H)=(L^2-H^2). I think especially the last part is quite difficult in getting the right sign. Any way thank you very much for your help!