12.9. Common Discrete Distributions and Their PMFs

Bernoulli(p)

$P(X = x) = p^x(1−p)^{1−x}, x ∈ {0,1}$

The Bernoulli distribution is a simple yet fundamental discrete distribution. It represents a single trial with two possible outcomes—success or failure. The parameter 'p' indicates the probability of success. When we say $P(X = 0)$, it means the event did not happen (failure), while $P(X = 1)$ means the event did occur (success). In essence, if you flip a coin, getting Heads could represent success.

Imagine you are tossing a coin where ‘Heads’ is a success (let's say you win a dollar) and ‘Tails’ is a failure (you win nothing). If the coin is fair, the probability ‘p’ for Heads is 0.5, and hence the probability of getting Tails will also be 0.5. Thus, this distribution helps model simple yes-or-no experiments.

Binomial(n, p)

$P(X = x) = \binom{n}{x} p^x(1−p)^{n−x}$

The Binomial distribution extends the Bernoulli distribution to multiple trials. It describes the number of successes in 'n' independent Bernoulli trials, each with the same probability of success 'p'. The term $\binom{n}{x}$ (read as 'n choose x') calculates how many ways 'x' successes can occur in 'n' trials. For instance, if you flip a coin 10 times, you could get 7 Heads in different arrangements, and the binomial formula helps calculate that probability.

Think about a basketball player who has a free throw success rate of 75% (or p = 0.75). If they take 10 shots, the Binomial distribution would help us find out the probability of scoring exactly 8 baskets. This distribution is important in many fields whenever there are a fixed number of trials.

Geometric(p)

$P(X = x) = (1−p)^{x−1}p, \text{ for } x = 1, 2, ...$

The Geometric distribution models the number of trials needed to achieve the first success in repeated independent Bernoulli trials. It shows that 'x' is the trial on which the first success occurs. For example, if you keep flipping a coin until you get your first Head, the number of flips required follows the Geometric distribution, which considers the probability of getting a Tail (failure) in the previous trials plus the success (Head) on the current trial.

Imagine you’re playing a game where you roll a die until you get a six. The Geometric distribution can tell you the likelihood that you will roll your first six on the third roll (after two non-sixes). It's particularly useful in scenarios where we want to know how long it will take to succeed.

Poisson(λ)

$P(X = x) = \frac{e^{−λ}λ^x}{x!}, \text{ for } x = 0, 1, 2,...$

The Poisson distribution describes the number of events that occur within a fixed interval of time or space, where these events occur with a known constant mean rate (λ) and are independent of the time since the last event. It's particularly useful for modeling rare events. For instance, if on average 3 customers arrive at a store every hour, the Poisson distribution can help calculate the probability that exactly 5 customers arrive in the next hour.

Consider a call center that receives an average of 5 calls per minute. The Poisson distribution can be applied to predict how many calls will be received in a given minute. If the average is 5 (λ), we can use this distribution to find out the probability of receiving 10 calls or none at all during the minute.

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