Kamil Wylegała — 2010 — May

Posts from May 2010

Law of the Unconscious Statistician

May 3rd, 2010

In polish it's called "Prawo leniwego statystyka" ("Law of the Unconscious Statistician") and it is a property of expected value of discrete random variable. It says:

If $\100dpi \inline X$ is an random variable with values $\100dpi \inline x_i$ and its probabilities $\100dpi \inline f(x_i)=P(X=x_i)$ for each $\100dpi \inline i>0$ then for any function $\100dpi \inline g:D\rightarrow\mathbb{R}$ :

$\100dpi E(g(X))=\sum_{i}g(x_i)f(x_i)=\sum_{i}g(x_i)P(X=x_i)$

Proof:

$\100dpi \sum_{i}g(x_i)f(x_i)=\sum_{j}\sum_{i:g(x_i)=y_j}g(x_i)f(x_i)=$

$\100dpi \sum_{j}y_j\sum_{i:g(x_i)=y_j}f(x_i)=\sum_{j}y_jP(g(X)=y_j)=E(g(X))=Eg(x)$ .

Example:

Find the expected value of variable $\100dpi \inline Y=(-1)^X$ , where $\100dpi \inline X$ has following distribution:

$\100dpi \inline P(X=0)=0,3 \quad P(X=1)=0,1 \quad P(X=3)=0,6$

Solution:

$\100dpi \inline EY=E[(-1)^X]=(-1)^0\cdot 0,3 + (-1)^1\cdot 0,1+(-1)^3\cdot 0,6=$

$\100dpi \inline 0,3-0,1-0,6=-0,4$ .

One comment...» Categories: Mathematics. Tags: examples, probability, theorems.