91Ó°ÊÓ

A particle starts at the origin of the whole line and moves along the line in jumps of one unit.

For every jump, the chance is p (0 ≤ p ≤ 1) of the particle jumping one unit to the left is p (0 p 1), as well as the probability of the particle jumping one unit to the right, is 1 – p.

X is the random variable that takes two values -1 with probability p and 1 with probability 1-p.

\(\begin{array}{c}E\left( X \right) = - 1 \times p + 1 \times \left( {1 - p} \right)\\ = 1 - 2p\end{array}\)

Step 3: Find the expected value of the position of the particle after n jumps

Let Y be the random variable where \(Y = {X_1}{X_2}...{X_n}\)\(\)

\(\begin{array}{c}Y = {X_1}{X_2}...{X_n}\\E\left( Y \right) = E\left( {{X_1}{X_2}...{X_n}} \right)\\\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {1 - 2p} \right)^n}\,{\rm{as}}\,{X_1},{X_2},...{X_n}\,{\rm{are}}\,{\rm{independent}}\end{array}\)