713. Binomial Random Variable - JulTob/Mathematics GitHub Wiki
A binomial random variable is based on an underlying Bernoulli experiment, meaning an event that can either occur (success) or not occur (failure). The binomial variable consists of repeating the Bernoulli experiment a certain number of times and recording the number of successes. Each repetition is independent of the others.
- Number of trials: n=20n = 20
- Probability of success per trial: p=0.0001p = 0.0001
Each value has a specific probability of occurrence:
| Variable value kk | Probability pX(k)p_X(k) | kpX(k)k p_X(k) | k2pX(k)k^2 p_X(k) |
|---|---|---|---|
| 0 | 0.9980018989 | 0 | 0 |
| 1 | 0.001996203418 | 0.001996203418 | 0.001996203418 |
| 2 | 0.000001896582905 | 0.000003793165811 | 0.000007586331622 |
| 3 | 0.00000000113806355 | 0.000000003414190649 | 0.00000001024257195 |
| 4-19 | 0 | 0 | 0 |
- The variable represents the number of successes in nn trials.
- Each trial is independent and has the same probability of success.
- The expectation is given by n⋅pn \cdot p.
- The variance is given by n⋅p⋅(1−p)n \cdot p \cdot (1 - p).
This type of random variable is widely used in:
- Modeling repeated binary outcomes, such as pass/fail tests or defect counts in manufacturing.
- Predicting the likelihood of a certain number of occurrences in genetics, finance, and risk assessment.
- Statistical inference, where binomial experiments provide the basis for hypothesis testing and confidence intervals.
---
config:
look: handDrawn
theme: dark
---
xychart-beta
title "L=10"
x-axis 0 --> 10
y-axis "Chances" 0 --> 1
bar [ 0.9, 0.09, 0.009, 0.0009, 0, 0, 0, 0, 0, 0]
line [0.9, 0.99, 0.999, 0.9999, 1, 1, 1, 1, 1, 1, 1, 1, 1]
