Solving for \pi_{H} and \pi_{L} in Limit Case

[kayhm]

Is there a way to solve for $$\pi_{H}$$ and $$\pi_{L}$$ which are probabilities when:

$$\pi_{H}$$ N$$_{H}$$ + $$\pi_{L}$$ N$$_{L}$$ = 1
(1 - $$\pi_{H}$$)$$^{M-1}$$ y$$_{H}$$ = k
(1 - $$\pi_{L}$$)$$^{M-1}$$ y$$_{L}$$ = k

It s ok to solve it for the limit case as N$$_{H}$$, N$$_{L}$$, and M go to infinity.

 

[sachinism]

leave alone solving it , i do not even understand the symbols in the question

can anyone explain to me what they are?

 

[Goongyae]

His notation is godawful but I think he wants to solve

AN^H + BN^L = 1
(1-A)^(M-1)Y^H = k
(1-B)^(M-1)Y^L = k

for A and B. And he's writing pi_h for A and pi_L for B

In this case the solution is obvious:
A =1 - (kY^-H)^1/(M-1)
B =1 - (kY^-L)^1/(M-1)
And the first equation must still be true meaning there is some kind of constraint on k, Y, H, M and L, and N whatever the heck those are.

 

[jhooper3581]

Given two equations don't look too hard to solve, so why not give it a try?

 

[kayhm]

Sorry. i fixed the notations to an easier to see format.

Is there a way to solve for A and B which are probabilities when:

AN + BL = 1
(1 - A)^(M-1) x = k
(1 - B)^(M-1) y = k

It s ok to solve it for the limit case as N, L, and M go to infinity.