If x is a normal variate with mean 10 and S.D. 4, find (i) P(|x-14|<1 (ii)P(5≤x≤18) (iii)P(x≤12).

Subject: Applied Mathematics 2

Topic: Probability Distribution

Difficulty: High

Z = $\frac{(X-m)}{σ}$ = $\frac{(X-10)}{4}$

(i) When X =14, Z = $\frac{(14-10)}{4}$ = 1

∴ P(|X-14|<1 = P(|Z|≤1) = area between (Z= - 1 & Z = 1)

= 2( area between Z=0 to Z= 1)

= 2(0.3413) = 0.6826

(ii) When X=5, Z = $\frac{(5-10)}{4}$ = - 1.25 & when X=18, Z = $\frac{(18-10)}{4}$ = 2

∴ P(5≤X≤18) = P(-1.25≤Z≤2)

=area between Z= - 1.25 to Z= 2

(area between Z =0 to Z = 1.25) + (area between Z =0 to Z = 2)

=0.3944 + 0.4772 = 0.8716

(iii) When X =12, Z = $\frac{(12-10)}{4}$ = 0.5

∴ P(X≤12) = P(Z≤0.5)

= (area from Z = - $\infty$ to Z= 0 ) + (area from Z=0 to Z=0.5) = 0.5 + 0.1915 = 0. 6915