Question

Basic Probability Let us consider a sequence of Bernoulli trials with probability of success p. Such a sequence is observed until the first success occurs. We denote by X the random variable (r.v.), which gives the trial number on which the first success occurs. This way, the probability mass function (pmf) is given by Px(x) = (1 – p)?-?p which means that will be x 1 failures before the occurrence of the first success at the x-th trial. The r.v. X defined as above, is called the geometric r.v. with parameter p. Solve the following tasks (Hint: use the sum of the geometric series Enzo ar” = {n-1 arn-1 for [r[ < 1) : 1-1 a (a) Prove that the geometric r.v. satisfies the total probability axiom, or in other words, 22-1Px(x) = 1.

Answer 1

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