The Archive

To see problems and solutions in the fall series, which runs from October 13 through December 15 visit The Fall 2025 Featured Problem Series

Problem

This week our problem comes from Penn State Math 414 an upper division probability course that does not use measure theory. Unlike many of the problems in this course, the random variable is not one of the commom named random variables. So the solution will take some thought.

Consider a sequence of independent Bernoulli trials with fixed probability of success . The trials will end after the first success is observed or after trials if no success has occurred by then. Let denote the number of trials conducted. Find .

Solution

Consider the related procedure, a sequence of independent Bernoulli trials with fixed probability of success terminates after the observation of the first success. A well-known result for this procedure is that the number of trials before termination is a geometric random variable with parameter . The solution of the given problem will proceed by expressing in terms of and exploiting known properties of geometric random variables.

To express in terms of a new new Bernoulli random variable is introduced. It is defined by

With this new random variable one may express in terms of as follows To exploit the fact that the geometric distribution is memoryless, this equation is rearranged to give By the memoryless property, or equivalently .

Now, is found. The linearity of expectation and the well-known fact that yields Conditioning on the value of is used to find : The second and third equalities in the above sequence of equalities follows from , and , respectively. The fourth equality is a consequence of the memoryless property. Finally, observe that . This is the event that it takes more than trials for the sequence of trials to terminate. which will occur if and only if the first trials are all failure. Thus by independence of the trials and constant probability of success, Equations ,, and are combined to give