Normally distributed random variable?

Well,

a continuously distributed probability (and the normal distribute is one !! ) a (positive) "measure" whose total "weight" is 1.

let X be the RV of distribution f_X(x)

for any measurable set J in the domain (here : R)

P(X € J) = ∫ (x € J) f_X(x) dx

typically : if J = (a,b) is an interval then it 's the integral from a to b

BUT :

the probability that a continuously distributed probability takes a precise value x0 is ZERO !!

BECAUSE :

{ x0 } 's measure is 0

or, if you prefer :

P(X = x0) = ∫ (x € { x0 } f_X(x) dx = 0

conclusion : P(X = 1000) = 0

hope it' ll help !!