Probability mass function of sum of two independent geometric random variables
How could it be proved that the probability mass function of , where X and Y are independent random variables each geometrically distributed with parameter p; i.e. equas to
Answer & Explanation
Since , the summation should run over .
Using this your convolution becomes
A geometric random variable is the count of Bernouli trial until a success. We measure the probability of obtaining failures and then 1 success.
The sum of two such is the count of Bernouli trials until the second success. We measure the probability of obtaining 1 success and failures, in any arrangement of those trials, followed by the second success.
This may also be counted by summing
Since must equal n and neither can be less than 1, then neither can be more than . Hence this the range of X values we must sum over.
Most Popular Questions