What is the expected distance E|X_n| for a simple random walk? I present a simple proof using a tricky martingale

This video inspired by this blog post https://angeris.github.io/blog/conten...
and this Tweet https://twitter.com/GuilleAngeris/sta...

See also my video on using martingales to prove the ABRACADABRA theorem for the expected time until a given sequence appears when flipping coins    • A Coin Flip Paradox and the ABRACADAB...  

Timestamps:
0:00 Expected value of |X_n| for simple random walk
0:34 The answer is E|X_n| = sqrt(2/pi) sqrt(n)
0:57 Upper bound E|X_n| less than sqrt(n)
2:00 What is a martingale?
2;30 The martingale for this problem
4:22 Why is it a martingale? Tricky sum
5:19 Discrete Tanaka Formula
5:50 Direct proof of why is it a martingale Case |X_n|=0
8:22 Case |X_n| is not zero
9:54 Using the martingale Optional Stopping Theorem
11:34 E|X_n| is a sum of probabilities to be at
12:15 Estimate Probabilities using the local Central Limit Theorem
13:50 Approximate sum as an integral