Probability Distribution


Let
x0 = 0
x1 = -1 with probability p
= 1 with probability 1 - p

x_(j+1) = xj - 1/2^j with probability p
= xj + 1/2^j with probability 1 - p

What is the accumulitive probability distribution of x = lim(j->oo) xj ?

When p = 1/2, the distribution is uniform and has value
p(x <= b) = 0, for b < -2
p(x <= b) = (b + 2)/4, for b in [-2,2]
p(x <= b) = 1, for 2 < b

When p = 0, lim_j xj = 2
p(x <= b) = 0, for b < 2
p(x <= b) = 1, for 2 <= b

When p = 1, lim_j xj = -2
p(x <= b) = 0, for b < -2
p(x <= b) = 1, for -2 <= b

What for other values of p is the nature of p(x < b)?

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