alamorge

02-18-2010, 12:14 PM

Suppose that N individuals have a certain gain "a" at t0 and that each of them at any point in time participate to an independent lottery which distribute a gain proportional to the initial gain with coefficient of proportionality b. The lottery is a Poisson with parameter L.

That is after the first lottery has been won, the distribution of gains is

a with frequency (N-1)/N

ba with frequency 1/N

Obviously at the second win it can be either of the following two cases

1.

a with frequency (N-1)/N

bba with frequency 1/N

2.

a with frequency (N-2)/N

ba with frequency 2/N

according to whether the first winner wins again or not.

After three wins things are as follows

1.

a with frequency (N-1)/N

bbba with frequency 1/N

2.

a with frequency (N-2)/N

ba with frequency 1/N

bba with frequency 1/N

3.

a with frequency (N-3)/N

ba with frequency 3/N

I am wondering which is the distribution of gains after a certain number of wins, say T, has occurred. Is that possible to recover which is its probability density function and cumulative density function?

Many thanks.

That is after the first lottery has been won, the distribution of gains is

a with frequency (N-1)/N

ba with frequency 1/N

Obviously at the second win it can be either of the following two cases

1.

a with frequency (N-1)/N

bba with frequency 1/N

2.

a with frequency (N-2)/N

ba with frequency 2/N

according to whether the first winner wins again or not.

After three wins things are as follows

1.

a with frequency (N-1)/N

bbba with frequency 1/N

2.

a with frequency (N-2)/N

ba with frequency 1/N

bba with frequency 1/N

3.

a with frequency (N-3)/N

ba with frequency 3/N

I am wondering which is the distribution of gains after a certain number of wins, say T, has occurred. Is that possible to recover which is its probability density function and cumulative density function?

Many thanks.