1

THE NORMAL DISTRIBUTION

In the analysis so far, we have discussed the mean and the variance of a distribution of a random variable, but we have not said anything specific about the actual shape of the distribution. It is time to do that.

00 m

f(X)

X

2

There are only four distributions, all of them continuous, that are going to be of importance to us: the normal distribution, the t distribution, the F distribution, and the chi-squared (2) distribution.

THE NORMAL DISTRIBUTION

00 m

f(X)

X

3

The normal distribution has the graceful, bell-shaped form shown.

THE NORMAL DISTRIBUTION

00 m

f(X)

X

4

The probability density function for a normally distributed random variable X is as shown,where and are parameters.

THE NORMAL DISTRIBUTION

00 m

f(X)

X

2

21

21

X

eXf

5

It is in fact an infinite family of distributions since can be any finite real number and any finite positive real number.

THE NORMAL DISTRIBUTION

00 m

f(X)

X

2

21

21

X

eXf

6

It can be shown that the expected value of the distribution, m, is equal to and its variance, 2, is equal to 2. Thus it is natural to write the probability density function as shown.

THE NORMAL DISTRIBUTION

00 m X

f(X)

m+ m+2 m+3m 3 m m 2m 4 m+4

2

21

21

m

X

eXf

7

The distribution is symmetric, so it automatically follows that the mean and the mode coincide in the middle of the distribution.

THE NORMAL DISTRIBUTION

00 m X

f(X)

m+ m+2 m+3m 3 m m 2m 4 m+4

2

21

21

m

X

eXf

8

The shape of the distribution is fixed when expressed in terms of standard deviations, so all normal distributions look the same when expressed in terms of m and . This is shown in figure.

THE NORMAL DISTRIBUTION

00 m X

f(X)

m+ m+2 m+3m 3 m m 2m 4 m+4

2

21

21

m

X

eXf

9

As a matter of mathematical shorthand, if a variable X is normally distributed with mean m and variance 2, this is written X ~ N(m, 2). (The symbol ~ means ‘is distributed as’). The first argument in the parentheses refers to the mean and the second to the variance.

THE NORMAL DISTRIBUTION

00 m X

f(X)

m+ m+2 m+3m 3 m m 2m 4 m+4

2

21

21

m

X

eXf

2,~ mNX

10

This, of course, is the general expression. If you had a specific normal distribution, you would replace the arguments with the actual numerical values.

THE NORMAL DISTRIBUTION

00 m X

f(X)

m+ m+2 m+3m 3 m m 2m 4 m+4

2

21

21

m

X

eXf

2,~ mNX

11

An important special case is the standardized normal distribution, where m = 0 and = 1. This is shown in the figure.

THE NORMAL DISTRIBUTION

0

0.1

0.2

0.3

0.4

0.5

-5 -3 -1 1 3 X

f(X) 2

21

21 XeXf

1,0~ NX

Copyright Christopher Dougherty 2012.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2012.10.31