Question

Suppose that X ∼ N(500, 1002), find

(a) P(X ≥ 500)

(b) P(X < 300)

(c) P(200 ≤ X ≤ 800)

(d) P(300 < X ≤ 400)

Answer 1

a)

$$P(X\geq500)=1-P(X<500)$$

$$=1-P(Z<\dfrac{500-500}{100})=1-P(Z<0)=0.5$$

b)

$$P(X<300)=P(Z<\dfrac{300-500}{100})=P(Z<-2)$$

$$=0.022750$$

c)

$$P(200\leq X\leq800)=P(X\leq800)-P(X<200)$$

$$=P(Z\leq\dfrac{800-500}{100})-P(Z<\dfrac{200-500}{100})$$

$$=P(Z\leq 3)-P(Z<-3)$$

$$=0.9986501-0.0013499=0.997300$$

d)

$$P(300<X\leq400)=P(X\leq400)-P(X\leq300)$$

$$=P(Z\leq\dfrac{400-500}{100})-P(Z\leq\dfrac{300-500}{100})$$

$$=P(Z\leq -1)-P(Z\leq-2)$$

$$=0.1586553-0.0227501=0.135905$$