Uncertainty/Examples/Example 1

From ProofWiki
Jump to navigation Jump to search

Example of Uncertainty

Let $R_1$ and $R_2$ be horseraces.


Let $R_1$ have $7$ runners:

$3$ of which each have probability $\dfrac 1 6$ of winning
$4$ of which each have probability $\dfrac 1 8$ of winning.


Let $R_2$ have $8$ runners:

$2$ of which each have probability $\dfrac 1 4$ of winning
$6$ of which each have probability $\dfrac 1 {12}$ of winning.


Then $R_1$ and $R_2$ have equal uncertainty.


Proof

Let $H_1$ and $H_2$ denote the uncertainty of $R_1$ and $R_2$ respectively.

Recall the definition of uncertainty:

$\map H X = \displaystyle -\sum_k p_k \lg p_k$

Thus:

\(\displaystyle H_1\) \(=\) \(\displaystyle -\paren {\sum_{k \mathop = 1}^3 \dfrac 1 6 \lg \dfrac 1 6 + \sum_{k \mathop = 1}^4 \dfrac 1 8 \lg \dfrac 1 8}\)
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac 1 6 \paren {3 \lg \dfrac 1 6} - \dfrac 1 8 \paren {4 \lg \dfrac 1 8}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac 1 2 \lg \dfrac 1 6 - \dfrac 1 2 \lg \dfrac 1 8\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 2 \paren {\lg 6 + \lg 8}\) Logarithm of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 2 \lg 48\) Sum of Logarithms


and:

\(\displaystyle H_2\) \(=\) \(\displaystyle -\paren {\sum_{k \mathop = 1}^2 \dfrac 1 4 \lg \dfrac 1 4 + \sum_{k \mathop = 1}^6 \dfrac 1 {12} \lg \dfrac 1 {12} }\)
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac 1 4 \paren {2 \lg \dfrac 1 4} - \dfrac 1 {12} \paren {6 \lg \dfrac 1 {12} }\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac 1 2 \lg \dfrac 1 4 - \dfrac 1 2 \lg \dfrac 1 {12}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 2 \paren {\lg 4 + \lg 12}\) Logarithm of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 2 \lg 48\) Sum of Logarithms

Hence the result.

$\blacksquare$


Sources