Question: This question has three parts.

1. The probability density function of the Gamma distribution is
   \[f_{X}(x ; \lambda, r)=\frac{\lambda}{\Gamma(r)}(\lambda x)^{r-1} e^{-\lambda x}, \quad x \geq 0\]
   Compute the generating function, \(G_{X}(s)\) of the Gamma distribution.
   Show all step for above question 10 (a)
2. The probability density function of the exponential distribution is
   \[f_{Y}(y ; \lambda)=\lambda e^{-\lambda y}, \quad x \geq 0\]
   Compute the generating function, \(G_{Y}(s)\) of the exponential distribution.
3. Using the generating functions establish a general relation between \(\Gamma \) random variables and exponential random variables.

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