Introduction to Probability · Problem 17

Expected Value and Variance — Grinstead & Snell

Mathematics — Probability · variance-discrete-random-variables

Problem

Let

X

be the outcome of a chance experiment with

E(X) = \mu

and

V(X) = \sigma^2

. When

\mu

and

\sigma^2

are unknown, the statistician often estimates them by repeating the experiment

n

times with outcomes

x_1, x_2, \ldots, x_n

, estimating

\mu

by the sample mean

\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i,

and

\sigma^2

by the sample variance

s^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2.

Then

s

is the sample standard deviation. (Many statisticians define the sample variance with the coefficient

1/n

replaced by

1/(n-1)

. If this alternative definition is used, the expected value of

s^2

is equal to

\sigma^2

. See Exercise 18, part (d).) Write a computer program that will roll a die

n

times and compute the sample mean and sample variance. Repeat this experiment several times for

n = 10

and

n = 1000

. How well do the sample mean and sample variance estimate the true mean

7/2

and variance

35/12