Binomial Trial

A binomial trial is a discrete simple random experiment from a binary sample space (typically labeled as success outcome and failure outcome).

• AKA: Bernoulli Random Experiment.
• Context:
  • It can be represented by a Binomial Random Variable (with success probability [math]\displaystyle{ p }[/math]).
  • It can be a component of a Binomial Process.
  • It can range from being a Binomial Trial with Equal Probability of Independent Events to being a Poisson Trial.
  • …
• Example(s):
  • 10 coin flips.
  • …
• Counter-Example(s):
  • a Multinomial Trial.
• See: Bernoulli Event, Binomial Mass Function, Bernoulli Probability Function, Random Experiment Trial.

---

---

References

2013

• (Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Bernoulli_trial
  • In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure". The mathematical formalization of the Bernoulli trial is known as the Bernoulli process. This article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment.

    In practice it refers to a single experiment which can have one of two possible outcomes. These events can be phrased into "yes or no" questions

2013

• http://en.wikipedia.org/wiki/Bernoulli_trial#Definition
  • Independent repeated trials of an experiment with two outcomes only are called Bernoulli trials.

    Call one of the outcomes "success" and the other outcome "failure".

    Let [math]\displaystyle{ p }[/math] be the probability of success in a Bernoulli trial. Then the probability of failure [math]\displaystyle{ q }[/math] is given by :[math]\displaystyle{ q = 1 - p }[/math]. Random variables describing Bernoulli trials are often encoded using the convention that 1 = "success", 0 = "failure".

2006

• (Dubnicka, 2006f) ⇒ Suzanne R. Dubnicka. (2006). “Special Discrete Distributions - Handout 6." Kansas State University, Introduction to Probability and Statistics I, STAT 510 - Fall 2006.
  • BERNOULLI TRIALS: Many experiments consist of a sequence of trials, where
    • (i) each trial results in a “success” or a “failure,”
    • (ii) there are n trials (where n is fixed),
    • (iii) the trials are independent, and
    • (iv) the probability of “success,” denoted by p, 0 < p < 1, is the same on every trial.
  • TERMINOLOGY : In a sequence of n Bernoulli trials, denote by X the number of successes (out of n). We call X a binomial random variable, and say that “X has a binomial distribution with parameters n and success probability p.” Shorthand notation is X ~ B(n, p).

2005

• (Lord et al., 2005) ⇒ Dominique Lord, Simon P. Washington, and John N. Ivan. (2005). “Poisson, Poisson-gamma and zero-inflated regression models of motor vehicle crashes: balancing statistical fit and theory.” In: Accident Analysis & Prevention, 37(1). doi:10.1016/j.aap.2004.02.004
  • QUOTE: A crash is, in theory, the result of a Bernoulli trial. Each time a vehicle enters an intersection, a highway segment, or any other type of entity (a trial) on a given transportation network, it will either crash or not crash. For purposes of consistency a crash is termed a “success” while failure to crash is a “failure”. For the Bernoulli trial, a random variable, defined as X, can be generated with the following probability model: if the outcome w is a particular event outcome (e.g. a crash), then [math]\displaystyle{ X(w)=1 }[/math] whereas if the outcome is a failure then [math]\displaystyle{ X(w)=0 }[/math]. Thus, the probability model becomes …

---