In: Statistics and Probability
Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k −1 trials are all failures and the kth a success. X is called a Geometric random variable (google it). Determine the moment generating function of X.
Independent trials, each of which is a success with probability p, are successively performed.
So, probability of failure is 1-p.
Let, X denote the first trial, resulting in a success.
ie. X will equal k, if the first k-1 trials are all failures, and the k th trial is a success.
So, the probability mass function of X is
Where, k can take any integer value greater than 0.
Now, we have to find the moment generating function of X.
Now, moment generating function of a random variable is given by
So, here the moment generating function will be
Now, the part under summation is an infinite geometric progression; so, by formula of sum of infinite geometric progression,
So, the moment generating function of X, which is a geometric random variable, with probability of success p, is