s********t
发帖数: 31 | 1 This is not the Gambler's ruin problem although relevant.
An object does random walk on a number line.
At time zero it is at location zero.
Probability to move to left (negative direction) is p. Here p>1/2
Probability to move to right (positive direction) is q=1-p.
Questions:
a. What is the expected number of steps for the object to get to -10?
b. What is the expected number of steps for the object to get to 10?
My thought for a is: Expected change per step (conditioned on the directino
of |
j*****4
发帖数: 292 | 2 a.10/(p-q)
b.infinity
use general gambler ruin result then take limiting case.
directino
【在 s********t 的大作中提到】
: This is not the Gambler's ruin problem although relevant.
: An object does random walk on a number line.
: At time zero it is at location zero.
: Probability to move to left (negative direction) is p. Here p>1/2
: Probability to move to right (positive direction) is q=1-p.
: Questions:
: a. What is the expected number of steps for the object to get to -10?
: b. What is the expected number of steps for the object to get to 10?
: My thought for a is: Expected change per step (conditioned on the directino
: of
|
s********t
发帖数: 31 | 3 jason04, thanks much.
I agree it is infinity for b (the same reason as in a). But it is a little
counter-intuitive as the object can hit 10 with good chance.
Also, could you or someone explain a little on case p=1/2? Take the
limiting p to 1/2?
【在 j*****4 的大作中提到】
: a.10/(p-q)
: b.infinity
: use general gambler ruin result then take limiting case.
:
: directino
|
p*****k
发帖数: 318 | 4 solarquant, if the prob of ever reaching 10 is strictly <1,
then there is nonzero prob of the first passage time being
infinity, hence the infinite average.
the nontrivial case is, as you said, when p=1/2. all points
on the line would be hit infinitely many times with prob of 1,
while the average first passage time is infinite.
one argument (which i personally feel misleading) is to consider
the well-known two-barrier version then take one of the barrier
to the infinity on the opposite side |
s********t
发帖数: 31 | 5 pcasnik, thanks for the insightful answer and a warm heart. |
