04 // Interactive Formula Compendium // STATISTICSH(X) = −∑ p_i · log₂(p_i), C = B · log₂(1 + S/N)
Shannon Information Entropy & Channel Capacity Formula
Binary Shannon entropy H(p) = −p·log₂(p) − (1−p)·log₂(1−p) measures expected surprise in bits per symbol, peaking at 1.000 bit when p = 0.50 and dropping to 0 bits for deterministic outcomes.
Formula Reference
Main Formula
H(X) = −∑ p_i · log₂(p_i), C = B · log₂(1 + S/N)
Rearranged Forms
H_binary(p) = −p·log₂(p) − (1−p)·log₂(1−p)
D_KL(P ∥ Q) = ∑ p_i · log₂(p_i / q_i)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| H | Binary Entropy H(p) | bits / symbol |
| p | Symbol Probability p | 0–1 |
| M | Equiprobable Alphabet Size M (info log₂M) | symbols |
Inputs & Parameters
[0–1]
[symbols]
Calculated Result (Binary Entropy H(p))
H =0.811278bits / symbol
Formula Notes & Background
- Claude Shannon (1948) proved H(X) = −∑ p_i log₂(p_i) is the unique continuous, symmetric, additive functional measuring uncertainty up to a logarithmic base factor.
- Base-2 logarithms yield 'bits' (shannons); natural logarithms yield 'nats' (1 nat = 1/ln2 ≈ 1.4427 bits).
Step-by-Step Examples
#1: Binary source emitting '1' with probability p = 0.25 and '0' with probability 0.75.
p = 0.251 − p = 0.75
H = −0.25·log₂(0.25) − 0.75·log₂(0.75) = 0.500 + 0.3113
→ H = 0.8113 bits/symbol (compresses ~18.9% below 1 raw bit/symbol)
#2: Shannon-Hartley capacity of a B = 20 MHz wireless channel at SNR = 31 (≈ 14.9 dB).
B = 20 MHzS/N = 31
C = 20 × 10⁶ × log₂(1 + 31) = 20 × 10⁶ × 5
→ C = 100 Mbps theoretical error-free limit
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Frequently Asked Questions
Why do we define 0 · log₂(0) = 0 in Shannon's formula?+
By L'Hôpital's rule, lim_{p→0⁺} p·log₂(p) = 0; impossible events contribute zero expected information.
When is discrete entropy H(X) maximized over M outcomes?+
Strictly when all M outcomes are equiprobable (p_i = 1/M), giving H_max = log₂(M) bits.
How does cross-entropy loss in machine learning relate to Shannon entropy?+
Cross-entropy H(P,Q) = H(P) + D_KL(P ∥ Q), so minimizing cross-entropy against true labels P directly minimizes Kullback-Leibler divergence D_KL.