Introduction to Probability · Problem 9.3.6

Generating Functions — Grinstead & Snell

Mathematics — Probability · central-limit-theorem-continuous

Problem

In general, the Central Limit Theorem gives a better estimate than Chebyshev's inequality for the average of a sum. To see this, let

A_{25}

be the average calculated in Exercise 5, and let

N

be the normal approximation for

A_{25}

. Modify your program in Exercise 5 to provide a table of the function

F(x) = P(|A_{25} - 10| \geq x) =

fraction of the total of 1000 trials for which

|A_{25} - 10| \geq x

. Do the same for the function

f(x) = P(|N - 10| \geq x)

. (You can use the normal table, Table 9.4, or the procedure NormalArea for this.) Now plot on the same axes the graphs of

F(x)

f(x)

, and the Chebyshev function

g(x) = 4/(3x^2)

. How do

f(x)

and

g(x)

compare as estimates for

F(x)