An infinite collection of lily-pads are arranged at integer points a line, and a
ID: 3249211 • Letter: A
Question
An infinite collection of lily-pads are arranged at integer points a line, and arc indexed by {..., -2. -1, 0, 1, 2, ...}. A frog is initially at lily-pad 0. At each time-step it hops one-step to either its left or right with equal probability. For example, suppose it is at lily-pad '-45'. Then at the next time-step, it can be at lily-pad '-44' or '-46'. After 10,000 time-steps, determine the expected distance that the frog has traveled from '0'. Further, using the central limit theorem, determine the probability that the frog is at-least a distance of 15 from '0' at time-step 10.000.Explanation / Answer
The expected distance after a long time will be zero since it can move left and right with equal probability
Probability that frog is at a distance of 15 from 0 is calculated as follows
Standard deviation = sqrt(n*4*p*(1-p)) where p=0.5
standard deviation = sqrt(10000*4*1/2*1/2) = 100
Z = (x-mean)/sigma = 15-0 / 100 = 0.15
P(z>0.15) can be found from normal tables as 1-P(z<0.15) = 1-0.559618=0.440382