Integrate Normal Distribution Between 2 Values-CodePudding

I want to calculate the integral of the Normal Distribution at exactly some point - I know that to do this, this is the equivalent of integrating the Normal Distribution at that point and at some point slightly after that point : then, you can subtract both of these values and get an approximate answer.

I tried doing this in R:

a = pnorm(1.96, mean = 0, sd = 1, log = FALSE)
b = pnorm(1.961, mean = 0, sd = 1, log = FALSE)

final_answer = b - a

 #5.83837e-05

Is it possible to do this in one step instead of manually subtracting "a" and "b"?

Thank you!

CodePudding user response：

We need to be clear about what you are asking here. If you are looking for the integral of a normal distribution at a specific point, then you can use pnorm, which is the anti-derivative of dnorm.

We can see this by reversing the process and looking at the derivative of pnorm to ensure it matches dnorm:

# Numerical approximation to derivative of pnorm:
delta <- 10^-6

(pnorm(0.75   delta) - pnorm(0.75)) / delta
#> [1] 0.3011373

Note that this is a very close approximation of dnorm

dnorm(0.75)
#> [1] 0.3011374

So the anti-derivative of a normal distribution density at point x is given by:

pnorm(x)

CodePudding user response：

You can try this

> diff(pnorm(c(1.96, 1.961), mean = 0, sd = 1, log = FALSE))
[1] 5.83837e-05