Independent trials, each of which is a success with probability p, are successively performed. Let X...

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.

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Expert Solution

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

orchestra answered 1 year ago

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