Question

What are the mean and standard deviation of the probability density function given by `(p(x))/k=sinx` for `x in [0,pi]`, in terms of k, with k being a constant such that the cumulative density across the range of x is equal to 1?

Answer 1

Mean: `mu = pi/2`
Standard Deviation: `sigma = 1/2sqrt(pi^2-8)`

Explanation:

The recipe is:

`color(red)"Step 1"`
Determine the probability distribution function pdf, p(x) such that:
`P(x) =int_(x_1)^(x_2)p(x) dx= 1 " over " [x_1, x_2]`
`1 = int_(0)^(pi)ksin(x)dx = 2k; k=1/2`
so your pdf is `p(x) = 1/2 sinx; [0, pi]`

`color(red)"Step 1"`
Determine the mean `mu` and standard deviation `sigma`
a) Mean: `mu = 1/2int_0^pixsinxdx = (sinx - xcosx)/2 = pi/2`
b) var: `sigma^2 = 1/2int_0^pi(x-mu)^2sinxdx`
`sigma^2 = [4(2x-pi)sinx-((2x-pi)^2 - 8)cosx]/2 = (pi^2-8)/4`
`sigma = 1/2sqrt(pi^2-8)`