Determine if what is underlined is true or false For the sim

Determine if what is underlined is true or false

For the simple symmetric random walk { X_n, n 0} with X_0 = 0 let us define the 1 : = inf { n 0: X_n = 1} and tho_ -1 : = inf { n 0: X_n = -1}. We have P(tho1 the -1) = 1/2.

Solution

Since it is a symettric random walk. there is an equal probability of body both the positiva and the negative step.

Hence,

P ( T₁ > T_-1 ) = P ( T₁ < T_-1 ) = 1/2

Since there is an equal chance of the next step to be ahead of the previous step or behind the previous step.