f*******e
发帖数: 1300 | 1 let's say X(i) is a random variable which equals K with a probability p and
2*K with a probability 1-p. S=X(1)+X(2)...+X(m) where m is the minimum
integer that makes S>=D and D is an integer too.the question is how I can
get an analytic formula of the expected number of m.
thanks for your help. | l******r
发帖数: 18699 | 2 wald equality
先证S是个martingal，然后m是个stopping time
then by wald equality you can get it
and
【在 f*******e 的大作中提到】
: let's say X(i) is a random variable which equals K with a probability p and
: 2*K with a probability 1-p. S=X(1)+X(2)...+X(m) where m is the minimum
: integer that makes S>=D and D is an integer too.the question is how I can
: get an analytic formula of the expected number of m.
: thanks for your help.
| n*****n
发帖数: 3123 | 3 m/(2-p), where mK is the smallest integer >= D, assume S0=0.
consider {Sn-nK(2-p)}, n>=0. Easy to show it is martingale. Then m and 0 are
two stopping time. use OST, you will get the result. |
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