Final answer:
The probability density function (pdf) of a gamma random variable integrates to one. To show this, you need to integrate the pdf over the entire range of the random variable, using the properties of the gamma function. The result of the integration should be one.
Step-by-step explanation:
The probability density function (pdf) of a gamma random variable integrates to one. This means that the total area under the curve of the pdf is equal to one.
To show this, we need to integrate the pdf over the entire range of the random variable. The pdf of a gamma random variable is given by:
f(x) = (1/Γ(k)θ^k) * x^(k-1) * e^(-x/θ)
where Γ(k) is the gamma function and θ is a positive constant.
To integrate this function and show that it evaluates to one, you can use the properties of the gamma function and perform the integration step by step.
After integrating, you should obtain a result of one, which proves that the pdf of a gamma random variable integrates to one.