# Thread: expected value of particle position

1. ## expected value of particle position

Suppose that a particle starts at the origin of the real line and moves along the line in jumps of one unit. For each jump, the probability is $\displaystyle p \ (0\leq p \leq1)$ that the particle will jump one unit to the left and the probability is $\displaystyle 1-p$ that the particle will jump one unit to the right. Find the expected value of the position of the particle after $\displaystyle n$ jumps.

2. ## Re: expected value of particle position

suppose $\displaystyle Y_i = 1$ if the particle jumps to the right, and $\displaystyle Y_i = -1$ if it jumps to the left..
so $\displaystyle P(Y_i = 1) = (1-p)\;and\; P(Y_i\;=\;-1)=p$
If $\displaystyle X_n$ is the position of the particle after n jumps, you have:
$\displaystyle X_n = \sum_{i=1}^n Y_i$

now find $\displaystyle E(Y_i) \;and\; then\; E(X_n)$

3. ## Re: expected value of particle position

$\displaystyle E(Y_i)=(1)(1-p)+(-1)(p)=1-2p$
$\displaystyle E(X_n)=\sum\limits_{i=1}^n E(Y_i)=\sum\limits_{i=1}^n 1-2p=n(1-2p)=n-2np$