Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

in text the formula for scaled random walk is:

W^(n) (t) = (1/√n) M_nt

in the example it says that:

set t=0.25, n=100 and consider the set of possible values of W^(100) (0.25) = 1/10 M_25. This random variable is generated by 25 coin tosses, and since the unscaled random walk M_25 can take the value of any odd integer between -25 and 25.My first question is that why unscaled random walk takes only the odd integer?

The scaled random walk W^(100) (0.25) can take any of the following values:

-2.5, -2.3, -2.1,......,-0.3,-0.1,0.1,0.3,.......,2.1,2.3,2.5

My second question is that why only this range?

In order for W^(100) (0.25) to take the value 0.1, we must get 13 heads and 12 tails in the 25 coin tosses. The probability of this is

P{W^(100) (0.25) = 0.1} = {(25!)/(13! *12!)} * (1/2)^25 = 0.1555

by drawing a histogram bar centered at 0.1 with area 0.1555, since this bar has width 0.2, its height must be 0.1555/0.2 = 0.7775.

My last question is that why 13 heads and 12 tails, and why we did the factorial part "{(25!)/(13! *12!)}"?

Thanks in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Limiting Distribution of Scaled Random Walk

**Physics Forums | Science Articles, Homework Help, Discussion**