Introduction to Probability · Problem 9.1.15

Central Limit Theorem — Grinstead & Snell

Mathematics — Probability · central-limit-theorem-bernoulli-trials

Problem

Recall that if

X

is a random variable, the cumulative distribution function of

X

is the function

F(x)

defined by

F(x) = P(X \leq x) .

(a) Let

S_n

be the number of successes in

n

Bernoulli trials with probability

p

for success. Write a program to plot the cumulative distribution for

S_n

. (b) Modify your program in (a) to plot the cumulative distribution

F^*_n(x)

of the standardized random variable

S^*_n = \frac{S_n - np}{\sqrt{npq}} .

N(x)

to be the area under the normal curve up to the value

x

. Modify your program in (b) to plot the normal distribution as well, and compare it with the cumulative distribution of

S^*_n

. Do this for

n = 10, 50

, and

100