Geometric distribution

The geometric distribution is a probability distribution that arises when we count the number $M$ of independent Bernoulli trials until the first occurrence of a success. $M$ is the geometric random variable, and is defined for ${\mathbb{Z}}_{n>0}$. The random variable roughly follows the trajectory of a geometric series.

The geometric RV is the only discrete RV that is memoryless in its trials.

Computations

The probability mass function of $M$ is given by:

$$P(M=k)=(1-p{)}^{k-1}p$$

where $k\in {\mathbb{Z}}_{n>0}$, and where $p=P(A)$ is the probability of “success” in each Bernoulli trial.

The probability that $M\le k$ is given by:

$$P(M\le k)=\sum _{j=1}^{k}p{q}^{j-1}=p\sum _{j^{\prime }=0}^{k-1}{q}^{j^{\prime }}=p\frac{1-q^k}{1-q}=1-q^k$$

If we’re interested in the number of failures before a success occurs (random variable $M^{\prime }$):

$$P(M^{\prime }=k)=(1-p{)}^{k}p$$

The expected value is given by:

$$E(M)=\frac{1}{p}$$

And the variance is given by:

$$\text{Var}(M)=\frac{1-p}{p^2}$$

Both increase as $p$ (the success probability) decreases.