Exercise 4: The exponential distribution is the continuous analogue of the geometric distribution. Show that, if X is exponentially distributed with parameter 1 > 0, then Y := [X], where [] is the floor (or greatest integer) function, is a geometrically distributed random variable with parameter p=1-1-(thus = - In(1 - p)) and taking values in the set {0,1,2,...}, i.e. the probability density function (PMF) of Y is given by the following expression: P(Y = k) = (1 - p)"p, k=0,1,2,3,... Note that in a sequence of independent trials, where a success occurs with probability p, instead of counting the number of trials before the first success, Y counts the number of failures before the first success.
