If each arrival is exponentially distributed, then the $k$th arrival time is Erlang distributed. The Erlang PDF is: $ f_{Y_k}(y) = \lambda e^{-\lambda y} \frac{(\lambda y)^{k-1}}{(k-1)!} $ How is this derived?
How is the Erlang pdf derived?
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0Here's a derivation given by convolution: http://www.math.unl.edu/~scohn1/428s05/queue3.pdf Essentially the Erlang distribution is the result of convolving the exponential distribution with itself k-1 times. Note that convolution as the sum of random variables is particularly important to understanding what's going on here. – 2016-10-05