Re: Probability of exceeding a specific value

On Sep 19, 2:27 pm, David Bernier <david...@xxxxxxxxxxxx> wrote:

[snip]

An interesting variation is the distribution (pdf) for the first time
a random walker returns to his starting place, on Z.
The expectation was said to be infinite. The formulation
was equivalent to a random walk on Z, but formulated as follows:
"Suppose we toss a fair coin. Let T ( positive) be first
toss number when we have equal numbers of heads and
tails." Clearly, T is even. It was mentioned that E(T) = oo.
I don't know the distribution of T, but the probability
of n heads and n tails in 2n tosses is C(2n, n)* 2^(-2n).
The pdf of T=2n is just A(2n)/(2^(2n)) , where
A(2n) is the number of random walk paths of length
2n steps where the starting point was visited only
once.

For this one it looks like the pdf of the first time to return to the
origin is, for even t,

C(t, t/2) / ((t - 1)*2^t)

and obviously zero for odd t.

It seems that you are certain to eventually return, but, as you say,
the expected time to do so is infinite.

.

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