why teh integral of density of probabilityfunction in normal distribution is a distribution function?

Question

i don't know calculate this integral..

Answer 1

This is true of any distribution that has a density, not just the normal one. It follows straight from the definitions. The c.d.f. (cumulative distribution function) of a vble X is by def. the fctn F(a) = Prob(X < a). Density, g, is the derivative of F: g(a) is by def. the limit of (1/epsilon)Prob(a < X < a+epsilon) as epsilon decreases to zero. So F(a) is the integral of g from -infty to a (this uses F(-infty) = 0 and the Fundamental Thm of Calculus, that integration reverses differentiation).

Answer 2

That's a definition. The integral of the density IS the cumulative function.

Doing that integral involves doing a double integral over polar coordinates: not for the newbie.