Consider a robot taking a random walk on the integer line. The robot starts at zero at time zero. After that, between any two consecutive integer times, the robot takes a unit length left step or right step, with each possibility having probability one half. Let F denote the event that the robot is at zero at time eight, and let X denote the location of the robot at time four (a) Find P(F).
(b) Find the pmf of X.
(c) Find P(X-i|F) for all integer values of i. (For what values of i is P(X-F) > 0?)
(d) Find the conditional pmf of X given that F is true. It is natural to use the notation px(ilF) for this, and it is defined by px(F) P(X-iF) for all integers i. Is the conditional pmf more spread out than the unconditional pmf px, or more concentrated?