Page 62 - Probability and Statistical Inference

1. Notions of Probability 39

C. F. Gauss, the celebrated German mathematician of the eighteenth cen-
tury, had discovered this distribution while analyzing the measurement errors
in astronomy. Hence, the normal distribution is alternatively called a Gaussian
distribution.

Figure 1.7.2. Normal Densities: (a) N(0,.25) (b) N(1,.25)

Let us ask ourselves: How can one directly check that f(x) given by (1.7.13)
is indeed a pdf? The function f(x) is obviously positive for all x ∈ ℜ. Next, we
need to verify directly that

Recall the gamma function Γ(.) defined by (1.6.19). Let us substitute u = (x –
2
µ)/σ, v = –u, w = 1/2v successively, and rewrite the integral from (1.7.14) as

In the last step in (1.7.15) since the two integrals are the same, we can claim
that