Month: February 2023

BUILDING INTUITION: FROM DISCRETE TO CONTINUOUS RANDOM VARIABLES

The most natural way to characterize a discrete distribution is by its PMF, which can be depicted as a set of bars whose height is the probability of each value. What happens when we consider a random variable that may take any real value on an interval? A starting point to build intuition is getting… Continue reading BUILDING INTUITION: FROM DISCRETE TO CONTINUOUS RANDOM VARIABLES

Introduction

We have gained the essential intuition about random variables in the discrete setting. There, we introduced ways to characterize the distribution of a random variable by its PMF and CDF, as well as its expected value and variance. Now we move on to the more challenging case of a continuous random variable. There are several… Continue reading Introduction

Poisson distribution

The Poisson random variable arises naturally when we have to count the number of events occurring over a specific time interval. We see that this kind of distribution is intimately related to exponential random variables, which are dealt with in Section 7.6.3, and with the Poisson stochastic process, introduced in Section 7.9. For now, the best way… Continue reading Poisson distribution

Binomial distribution

The binomial distribution arises as yet another variation on Bernoulli trials. We run n independent and identical experiments and let X be a random variable counting the number of successes. The support of the resulting random variable is {1, 2,…, n}, and its probability distribution depends on two parameters: the probability of success p and the number of experiments n. Since events are independent, it… Continue reading Binomial distribution

Geometric distribution

The geometric distribution is a generalization of the Bernoulli random variable. The underlying conceptual mechanism is the same, but the idea now is repeating identical and independent Bernoulli trials until we get the first success. The number of experiments needed to stop the sequence is a random variable X, with unbounded support 1, 2, 3,…. Finding its PMF… Continue reading Geometric distribution

Bernoulli distribution

The Bernoulli distribution is based on the idea of carrying out a random experiment, which may result in a success or a failure. Let p be the probability of success; then, 1 − p is the probability of failure. If we assign the value 1 to variable X in case of success, and 0 otherwise, we get the following PMF: It… Continue reading Bernoulli distribution

Discrete uniform distribution

The uniform distribution is arguably the simplest model of uncertainty, as it assigns the same probability to each outcome: This makes sense only if there is a finite number n of possible values that the random variable can assume. If they are consecutive integer numbers, we have an integer uniform distribution, which is characterized by the lower… Continue reading Discrete uniform distribution

Empirical distributions

Empirical distributions feature the closest link with descriptive statistics, since their PMF is typically estimated by collecting empirical relative frequencies. For instance, if we consider a sample of 10 observations of a random variable X, and X = 1 occurs in three cases, X = 2 in five cases, and X = 3 occurs twice, we may estimate Empirical distributions feature the largest… Continue reading Empirical distributions

A FEW USEFUL DISCRETE DISTRIBUTIONS

There is a wide family of discrete probability distributions, which we cannot cover exhaustively. Nevertheless, we may get acquainted with the essential ones, which will be illustrated by a few examples. First, we should draw the line between empirical and theoretical distributions. Since these terms may be a tad misleading, it is important to clarify… Continue reading A FEW USEFUL DISCRETE DISTRIBUTIONS