Bernoulli Process as Geometric Distribution/Shifted

Theorem

Let $\sequence {Y_i}$ be a binomial experiment with parameter $p$.

Let $\EE$ be the experiment which consists of performing the Bernoulli trial $Y_i$ as many times as it takes to achieve a success, and then stop.

Let $k$ be the number of Bernoulli trials to achieve a success.

Then $k$ is modelled by a shifted geometric distribution with parameter $p$.


Proof

Follows directly from the definition of shifted geometric distribution.

Let $Y$ be the discrete random variable defined as the number of trials for the first success to be achieved.

Thus the last trial (and the last trial only) will be a success, and the others will be failures.

The probability that $k-1$ failures are followed by a success is:

$\map \Pr {Y = k} = \paren {1 - p}^{k - 1} p$

Hence the result.

$\blacksquare$


Sources

  • 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.2$: Examples: Example $14$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): geometric distribution
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): geometric distribution
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Bernoulli trial
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bernoulli trial
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