The Normal Distribution

Several univariate continuous distributions that play a key role in statistical and econometric inference will be reviewed in this section, starting with the normal distribution. The standard normal distribution emerges as a limiting distribution ofan aggregate of random variables. In particular, if X1,Xn are independent random variables with expectation ц and finite and positive variance a2, then for large n the random variable Yn = (lUfn)YTj=1(Xj — l^)/a is approximately standard normally distributed. This result, known as the central limit theorem, will be derived in Chapter 6 and carries over to various types of dependent random variables (see Chapter 7).

4.5.1. The Standard Normal Distribution

The standard normal distribution is an absolutely continuous distribution with density function

Подпись: (4.28)„ ч exp(—x2/2) m

f(x) =———– ——, X є R.

J2n

image240

image241

which exists for all t є К, and its characteristic function is

tyN(0,i)(t) = m(i ■ t) = exp(—t2/2).

Consequently, if X is standard normally distributed, then

E[X] = m'(t)|t =0 = 0, E[X2] = var(X) = m"(t)|t =0 = 1.

Given this result, the standard normal distribution is denoted by N(0, 1), where the first number is the expectation and the second number is the variance, and the statement “X is standard normally distributed” is usually abbreviated as “X – N(0, 1).”

4.5.2. The General Normal Distribution

Подпись: f(x) Подпись: exp (— 2(x — Д)2 /a2) a V2n Подпись: x є К

Now let Y = д + aX, where X — N(0, 1). It is left as an easy exercise to verify that the density of Y takes the form

with corresponding moment-generating function

mN(JX, a2)(t) = E[exp(t ■ Y)] = exp(^t)exp(a2t2/2), t є К

and characteristic function

VN(v, a2)(t) = E[exp(i ■ t ■ Y)] = exp(i ■ ixt)exp(—a212/2).

Consequently, E[Y] = д, var (Y) = a2. This distribution is the general normal distribution, which is denoted by N(д, a2). Thus, Y — N(д, a2).

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