At each time step, I have 1/2 probability of walking one step to the right, and the same probability of walking one step to the left.
Let X be the random variable corresponding to the final position of the $n$ step I walk.
Compute
a) $E[X^4]$ for this random variable
b) Show that $P(|X|>c) \le \dfrac{E[X^4]}{c^4}$
My thought:
I tried to use the definition of expectation to compute but the polynomial of degree 4 got really messy. I was wondering if there is an elegant way to approach this problem. And I also tried to use generating function but how to write the generating function for this random variable.