Question

How can you write the geometric distribution into an exponential distribution?

For the geometric distribution: P(y; p) = p(1 − p) ^ y−1

for y = 1,2, ⋯ and 0 < p < 1.

Answer 1

The exponential distribution may be viewed as a continuous counterpart of the geometric distribution.

For the geometric distribution: for y = 1,2, ⋯ and 0 < p < 1.

The CDF of geometric distribution is,

for y = 1,2, ⋯ and 0 < p < 1.

Let the interval of y be divided into n equal intervals of length t. Then y = nt

Let the number of occurrence of event be in n time interval. Then p = /n

For continuous distribution,  

which is the CDF of exponential distribution.

Using the below limit formula,

Thus, the continuous distribution of geometric distribution is exponential distribution