712. Bernoulli Random Variable - JulTob/Mathematics GitHub Wiki

Definition

A Bernoulli random variable is one that describes an experiment with two possible outcomes: success or failure. The associated random variable can only take two values: 0 or 1, true or false, A or B, etc. Typically, 1 represents success and 0 represents failure.

Parameter

Probability of success: p=0.1p = 0.1

Probability Function

Each value has a specific probability of occurrence:

Variable value $k$	Probability $p_X(k)$	$k · p_X(k)$	$k^2 p_X(k)$
0	0.1	0.1	0.1
1	0.9	0	0

---
config:
  look: handDrawn
  theme: dark
---

xychart-beta
    title "L=10"
    x-axis [0,1]
    y-axis "Chances" 0 --> 1
    bar [0.1, 0.9]
    line [0.1, 1]

Properties

The variable takes only two values (binary outcome).
The probability of success is constant.
The expectation is equal to the probability of success.
The variance is given by p(1−p)p(1 - p), which depends on the probability of success.

Applications

This type of random variable is widely used in:

Binary decision modeling, such as pass/fail, win/loss scenarios.
Probability modeling in finance, health studies, and quality control.
Bernoulli trials, which are the basis of more complex distributions like the Binomial distribution.