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probabilitygamma-distribution

Gamma distribution

Let X~gamma(2,1).

Find:

P(X>=2)
(X<=10)

I'm struggling to determine how to solve this. I know you must integrate some function from 2 to infinity and 0 to 10 but don't know what to integrate.

Solution

  • There are actually two ways to define Gamma-distribution - one with scaling 'theta' and another with inverse scale 'beta', see https://en.wikipedia.org/wiki/Gamma_distribution

    Fortunately, for parameter value of 1 there is no difference.

    So you have to integrate PDF

    f(x) = (1/G(2)) * x * exp(-x)

    Gamma function at 2 is equal to 1. G(2)=1, so 

    P(x) = S x exp(-x) dx

    where S is integration notation.

    P(x) = -(x+1)*exp(-x)

    So now you have to substitute limits