The Bernoulli distribution is a discrete probability distribution that models the outcome of a single binary (yes/no, success/failure) experiment. It is characterized by a single parameter, the probability of success in a single trial.
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The Bernoulli distribution has only two possible outcomes: success (1) or failure (0).
The probability of success in a single Bernoulli trial is denoted by the parameter $p$, where $0 \leq p \leq 1$.
The expected value (mean) of a Bernoulli random variable is $p$, and the variance is $p(1-p)$.
Bernoulli trials are independent if the outcome of one trial does not affect the outcome of another trial.
Bernoulli trials are mutually exclusive if the occurrence of one outcome (success or failure) precludes the occurrence of the other outcome.
Review Questions
Explain how the Bernoulli distribution is related to the concepts of independent and mutually exclusive events.
The Bernoulli distribution is closely tied to the concepts of independent and mutually exclusive events. In a Bernoulli trial, the two possible outcomes (success or failure) are mutually exclusive, as the occurrence of one outcome precludes the occurrence of the other. Additionally, Bernoulli trials are independent, meaning the outcome of one trial does not affect the outcome of another trial. This independence is a key assumption of the Bernoulli distribution, which models the probability of success or failure in a single binary experiment.
Describe the relationship between the Bernoulli distribution and the binomial distribution.
The Bernoulli distribution is the foundation for the binomial distribution. The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure). The Bernoulli distribution is used to describe the probability of success or failure in a single trial, while the binomial distribution is used to describe the probability of a certain number of successes in a fixed number of independent Bernoulli trials. The binomial distribution can be seen as the sum of multiple Bernoulli trials.
Analyze how the parameters of the Bernoulli distribution, specifically the probability of success ($p$), impact the distribution's characteristics and applications.
The parameter $p$ in the Bernoulli distribution represents the probability of success in a single trial. This parameter directly influences the distribution's characteristics and applications. When $p$ is close to 0, the distribution models events that are unlikely to occur, while when $p$ is close to 1, the distribution models events that are highly likely to occur. The expected value (mean) of the Bernoulli distribution is equal to $p$, and the variance is $p(1-p)$, indicating that the distribution's spread is dependent on the value of $p$. The Bernoulli distribution is widely used in various fields, such as quality control, risk assessment, and decision-making, where the probability of a binary outcome is of interest.
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure).