(Solved): The special case of the gamma distribution in which \( \alpha \) is a positive integer \( n \) is ...

The special case of the gamma distribution in which \( \alpha \) is a positive integer \( n \) is called an Erlang distribution. If we replace \( \beta \) by \( \frac{1}{\lambda} \) in the expression below, \[ f(x ; \alpha, \beta)=\left\{\begin{array}{cc} \frac{1}{\beta^{\alpha} \Gamma(\alpha)} x^{\alpha-1} e^{-x / \beta} & x \geq 0 \\ 0 & \text { otherwise } \end{array}\right. \] the Erlang pdf is as follows. \[ f(x ; \lambda, n)=\left\{\begin{array}{rl} \frac{\lambda(\lambda x)^{n-1} e^{-\lambda x}}{(n-1) !} & x \geq 0 \\ 0 & x<0 \end{array}\right. \] (a) What is the expected value of \( X \) ? \[ E(X)= \] If the time (in minutes) between arrivals of successive customers is exponentially distributed with \( \lambda=0.5 \), how much time can be expected to elapse before the sixth customer arrives? \( \min \)