Examples

Markov and Chebyshev

A tire shop has 500 customers per day on average.

(a) Estimate the odds that more than 700 customers arrive today.

(b) Assume the variance in daily customers is 10. Repeat (a) with this information.

Solution

Write for the number of daily customers.

(a) Using Markov’s inequality with , we have:

(b) Using Chebyshev’s inequality with , we have:

The Chebyshev estimate is much smaller!

LLN: Average winnings

A roulette player bets as follows: he wins $100 with probability 0.48 and loses $100 with probability 0.52. The expected winnings after a single round is therefore 100\cdot 0.48 - $100\cdot 0.52-$4$.

By the LLN, if the player plays repeatedly for a long time, he expects to lose 4$ per round on average.

The ‘expects’ in the last sentence means: the PMF of the cumulative average winnings approaches this PMF:

This is by contrast to the ‘expects’ of expected value: the probability of achieving the expected value (or something near) may be low or zero! For example, a single round of this game.

Enough samples

Suppose are IID samples of .

(a) Compute and and .

(b) Use the finite LLN to find such that:

(c) How many samples are needed that to guarantee that: