Normal distribution Probabilities

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

Normal Distribution formulas


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`

See also

Inverse normal distribution Calculator
Statistics Calculators