SOME SPECIFIC PROBABILITY DISTRIBUTIONS Pr(a≤x≤b)=Pr(a-≤x-≤b-p) 0.1)-F(=0. area below 4. Probability of Intervals 3 2 b 1.96 a=1.6 We can then merely look in tables for the distribution function of a N(0, 1) variable 1.5. Moment generating function of a normal random variable. The moment generating function for the central moments is as follows Mx(t) The first central moment is E(Xx-p)=#(=)h=0 The second central moment is4 SOME SPECIFIC PROBABILITY DISTRIBUTIONS Consequently, P r(a ≤ x ≤ b) = P r (a − µ ≤ x − µ ≤ b − µ) = P r h a − µ σ ≤ x − µ σ ≤ b − µ σ i = F b − µ σ ; 0, 1  − F ￾ a − µ σ ; 0, 1
= area below (4) Figure 4. Probability of Intervals b - Μ €€€€€€€€€€€€€€€€€ Σ a - Μ €€€€€€€€€€€€€€€€ Σ .1 .2 .3 Μ = 0, Σ = 1 b = -1.96 a = 1.6 We can then merely look in tables for the distribution function of a N(0,1) variable. 1.5. Moment generating function of a normal random variable. The moment generating function for the central moments is as follows MX (t) = e t2 σ2 2 . (5) The first central moment is E (X − µ ) = d dt  e t2 σ2 2  |t = 0 = t σ2  e t2 σ2 2  |t = 0 = 0 (6) The second central moment is