A drunken man walks on a straight lane. At every integer time (in seconds) he moves a distance of $1$ unit randomly, either forwards or backwards. What is the expectation of the square of the distance after $100$ seconds from the initial position? Hint: The position at time $100$ is a sum of independent and identically distributed random variables.
- $100$
- $\frac{\sqrt{300}}{4}$
- $40$
- $200$
- $20 \pi$