QuantPrep
DashboardLearning PathsAdvanced StatisticsStatistical TestsCommunityAll Topics
Login
QuantPrep

© 2025 QuantfianceLab. All rights reserved.

Privacy PolicyTerms of ServiceContact
TwitterGitHubLinkedIn

    Bernoulli Distribution

    The fundamental building block of discrete probability, modeling a single trial with two outcomes.

    The "Single Coin Flip"

    The Bernoulli distribution is the simplest of all discrete distributions. It models a single event or trial that has only two possible outcomes: a "success" or a "failure".

    Think of it as a single coin flip (Heads or Tails), a single trade (Win or Loss), or a single bond (Default or No Default). The entire distribution is described by a single parameter, ppp, which is the probability of success.

    The Formula
    The probability mass function (PMF) is:
    P(X=k)=pk(1−p)1−kfor k∈{0,1}P(X=k) = p^k (1-p)^{1-k} \quad \text{for } k \in \{0, 1\}P(X=k)=pk(1−p)1−kfor k∈{0,1}
    • If k=1k=1k=1 (success), the formula becomes ppp.
    • If k=0k=0k=0 (failure), the formula becomes 1−p1-p1−p.
    Interactive Bernoulli Trial
    Adjust the probability of success (ppp) to see how it affects the outcome probabilities.