When studying sample spaces, we observe that they can be of two types: discrete and continuous. When the sample space is continuous, it is possible to define random variables of this nature and, from them, establish discrete probability distributions. We have already reviewed random variables here, now we will focus on discrete probability distributions.

The Concept of a Discrete Probability Distribution

We say that a random variable X has a discrete probability distribution if there exists a set C\subset\mathbb{R} finite or countably infinite such that P\left(X\in C\right)=1; thus, if we have values x\in C such that p_X(x) = P(X=x), it can be verified that if A\subset\mathbb{R}, then:

\begin{array}{lr} (*) & P\left(X\in A\right) = \displaystyle \sum_{x\in A \cap C} p_X(x) \end{array}

And in particular,

\begin{array}{lr} (**) & \displaystyle \sum_{x\in C} p_X(x) = 1. \end{array}

If we calculate P(X\in A) using A=]-\infty, t], we find that:

P(X\in A) = P(X\leq t) = F_X(t) = \displaystyle \sum_{x\leq t}p_X(x)

From this calculation, we conclude that F_X is a “step function” with jumps at x\in C of size p_X(x). The function p_X that goes from C to [0,1] is what we call the frequency function. Thus, a discrete distribution is given by a finite or countably infinite set C\subset \mathbb{R} and a function p_X(x)\geq 0 defined for each x\in C that satisfies expressions (*) and (**).

The 5 Most Well-Known Discrete Probability Distributions

In this section, we will continue our study on discrete probability distributions. We will now examine the 5 most well-known discrete probability distributions, illustrating the types of problems they can help solve.

Binomial or Bernoulli Distribution

The Binomial, or Bernoulli distribution, considers the random variable as the number of successes or failures (X) in n attempts with individual probability p. It is said that the random variable X follows a binomial distribution, X\sim Bi(n,p), then,

\displaystyle \large P(X=k)= {{n}\choose{k}} p^k(1-p)^{n-k}

Poisson Distribution

The Poisson processes are divided into two categories: spatial and temporal. This distinction arises from the decomposition of the parameter \lambda:

• Temporal case: \lambda=f\cdot T, where f is a frequency and T a time period.
• Spatial case: \lambda=\rho \cdot V, where \rho is a density and V a sample volume.

It is important to note that in both cases the parameter \lambda must be dimensionless. It should also be remembered that the Poisson process is a limiting case of the binomial process, so the random variable associated with this process is also linked to a certain “number of successes or failures.” It is said that the random variable X follows a Poisson distribution, X\sim Po(\lambda), if the following holds:

\large\displaystyle P(X=k)=\frac{\lambda^k}{k!}e^{-\lambda}

Geometric Distribution

Imagine a binomial process (such as repeatedly tossing a coin). If instead of asking about the number of successes after a certain number of attempts, you ask about the number of attempts you must make to obtain the first success, then you are dealing with a discrete random variable with a geometric distribution. A random variable X has a geometric distribution, X\sim Ge(p), if:

\displaystyle \large P(X=k)=p(1-p)^{k-1}

Negative Binomial Distribution

Similar to the Geometric is the Negative Binomial Distribution, but it is a bit more general. When you perform a binomial process (such as repeatedly tossing a coin) and instead of asking about the number of successes you ask about the number of attempts you make to obtain the m-th success, then you are dealing with a discrete random variable with a Negative Binomial distribution. If a random variable X has a Negative Binomial distribution, X\sim Bn(m,p), then:

\displaystyle\large P(X=k)= {{k-1}\choose{m-1}} p^m(1-p)^{k-m}

Hypergeometric Distribution

Imagine you have a bag with N colored spheres, of which M are white and the rest are black. If you draw n spheres from this bag (without replacement), then the number of white spheres drawn will be associated with a discrete random variable with a Hypergeometric distribution. If a random variable X has a Hypergeometric distribution, X\sim Hg(N,M,n), then:

\displaystyle \large P(X=k)=\frac{{{M}\choose{k}} {{N-M}\choose{n-k}}}{{N}\choose{n}}

Proposed Exercises

1. A board game store sells cards randomly from a lot of 500 collectible cards (imagine they are myth, magic, Pokemon, or any other TCG cards). If the seller ensures that there are always 450 common (low-value) cards and 50 rare (high-value) cards, what is the probability of obtaining 3 rare cards when buying 20 cards at random?

2. Using the following card in a game:

   

   What is the probability of discarding 4 of your opponent’s cards?

3. In a certain store, the probability of selling a device with a factory defect is 2%. What is the probability that the tenth device sold is the third with factory defects?