# Normal distribution Probabilities

Normal distribution Calculator : probabilities, density function (PDF) and cumulative density function (CDF).

## Normal Distribution formulas

### Notations

X : a random variable with a normal distribution
mu : mean of variable X
sigma : its Standard deviation

### Probability Density Function (PDF)

f(x) = 1/(sigma*sqrt(2pi))*e^(-1/2*(x-mu)^2/sigma^2)

### Cumulative distribution function (CDF)

F(x) = \int_-oo^x f(t)\ dt

F(x) = 1/(sigma*sqrt(2pi))*\int_-oo^xe^(-1/2*(t-mu)^2/sigma^2)\ dt

### Probabilities under Normal distribution

P(X < b) = F(b)

P(X > a) = 1 - F(a)

P(a < X < b) = F(b) - F(a)

We get the following formulas,

P(X < b) = \int_-oo^b f(t)\ dt = \int_-oo^b1/(sigma*sqrt(2pi))*e^(-1/2*(t-mu)^2/sigma^2)\ dt

P(X > a) = \int_a^{+oo} f(t)\ dt = \int_a^{+oo}1/(sigma*sqrt(2pi))*e^(-1/2*(t-mu)^2/sigma^2)\ dt

P(a < X < b) = \int_a^b f(t)\ dt = \int_a^b1/(sigma*sqrt(2pi))*e^(-1/2*(t-mu)^2/sigma^2)\ dt

Graphically, if we plot the curve of the density function f(x), these three probabilities are equal to the surface areas between the curve f(x) and x-axis, as shown in the diagram at the top of the page.

We have then,

P(X < a) + P(a < X < b) + P(X > b) = 1