Bernoulli Trial

Examples of Bernoulli Trial in the following topics:

• Binomial Probability Distributions

  • This chapter explores Bernoulli experiments and the probability distributions of binomial random variables.
  • Such a success/failure experiment is also called a Bernoulli experiment, or Bernoulli trial; when $n=1$, the Bernoulli distribution is a binomial distribution.
  • In a sequence of Bernoulli trials, we are often interested in the total number of successes and not in the order of their occurrence.
  • If we let the random variable $X$ equal the number of observed successes in $n$ Bernoulli trials, the possible values of $X$ are $0, 1, 2, \dots, n$.
  • Since the trials are independent and since the probabilities of success and failure on each trial are, respectively, $p$ and $q=1-p$, the probability of each of these ways is $p^x(1-p)^{n-x}$.

• Categorical Data and the Multinomial Experiment

  • The binomial distribution is the probability distribution of the number of successes for one of just two categories in $n$ independent Bernoulli trials, with the same probability of success on each trial.
  • In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number $k$ of possible outcomes, with probabilities $p_1, \cdots , p_k$ (so that $p_i \geq 0$ for $i = 1, \cdots, k$ and the sum is $1$), and there are $n$ independent trials.
  • Then if the random variables Xi indicate the number of times outcome number $i$ is observed over the $n$ trials, the vector $X = (X_1, \cdots , X_k)$ follows a multinomial distribution with parameters $n$ and $p$, where $p = (p_1, \cdots , p_k)$.
  • There are $k$ possible outcomes for each trial.
  • The probabilities of the $k$ outcomes, denoted by $p_1$, $p_2$, $\cdots$, $p_k$, remain the same from trial to trial, and they sum to one.

• Additional Properties of the Binomial Distribution

  • The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non-identical Bernoulli trials Bern(pi).
  • Using the definition of covariance, in the case n = 1 (thus being Bernoulli trials) we have .
  • Defining pB as the probability of both happening at the same time, this gives and for n independent pairwise trials .

• Binomial

  • There are a fixed number of trials.
  • Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial.
  • Any experiment that has characteristics 2 and 3 and where n = 1 is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively).
  • A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.
  • The number of trial is n = 50.

• The Hypergeometric Random Variable

  • A hypergeometric random variable is a discrete random variable characterized by a fixed number of trials with differing probabilities of success.

• Geometric distribution exercises

  • Determine if each trial can be considered an independent Bernouilli trial for the following situations.
  • The Bernoulli distribution allows for only two events or categories.
  • Note that rolling a die could be a Bernoulli trial if we simply to two events, e.g. rolling a 6 and not rolling a 6, though specifying such details would be necessary.
  • (e) When p is smaller, the event is rarer, meaning the expected number of trials before a success and the standard deviation of the waiting time are higher.

• Bernoulli distribution

  • Each person in Milgram's experiment can be thought of as a trial.
  • When an indi- vidual trial only has two possible outcomes, it is called a Bernoulli random variable.
  • A Bernoulli random variable has exactly two possible outcomes.
  • Bernoulli random variables are often denoted as 1 for a success and 0 for a failure.
  • Suppose we observe ten trials: 0 1 1 1 1 0 1 1 0 0

• Distributions of random variables solutions

  • The Bernoulli distribution allows for only two events or categories.
  • Note that rolling a die could be a Bernoulli trial if we simply to two events, e.g. rolling a 6 and not rolling a 6, though specifying such details would be neces- sary.
  • The conditions are satisfied: independence, fixed number of trials, either success or failure for each trial, and probability of success being constant across trials.
  • The negative binomial setting is appropriate since the last trial is fixed but the order of the first 3 trials is unknown.
  • In the negative binomial model the last trial is fixed.

• Introduction to geometric distribution (special topic)

  • We first formalize each trial – such as a single coin flip or die toss – using the Bernoulli distribution, and then we combine these with our tools from probability (Chapter 2) to construct the geometric distribution.

• The binomial distribution

  • The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with probability of a success p (in Example 3.37, n = 4, k = 1, p = 0.35).
  • Each trial outcome can be classified as a success or failure.
  • The probability of a success, p, is the same for each trial.
  • The number of trials is fixed (n = 8) (condition 2) and each trial outcome can be classified as a success or failure (condition 3).
  • (ii) We have a fixed number of trials (n = 4).