Chebyshev's Inequality

Tue, 05/18/2010 - 21:20 | kolier 0 comments
In probability theory, Chebyshev's inequality (also known as Tchebysheff's inequality, Chebyshev's theorem, or the Bienaymé–Chebyshev inequality) is one of the basic inequality.

Very important, with this base, some famous indicator spread widely much like Bollinger Bands and CCI.

Formula

<br /> \begin{equation}<br /> P(|X - E(X)| \geq k\cdot Dev(X)) \leq \frac{1}{k^2}<br /> \end{equation}<br /> Other Formats:<br /> \begin{equation}<br /> P(|X - E(X)|) \geq \epsilon) \leq \frac{Var(X)}{\epsilon^2} \label{Chebyshev's Inequality}<br /> \end{equation}<br /> or<br /> \begin{equation}<br /> P(|X - E(X)|) < \epsilon) \geq 1 - \frac{Var(X)}{\epsilon^2} \label{Chebyshev's Inequality}<br /> \end{equation}<br />
While

  • X is the random variable
  • E(X) is the expectation of the X
  • Dev(X) is the deviation of the X
  • Var(X) is the variance of X
  • any real number k > 0
  • any real number $ \epsilon $>0
  • In order to have a usable information, k and $ \epsilon $ have to be more than 1.

Where Could We Apply with Chebyshev's Inequality

This theory can apply to both Discrete and Continuous Random Variables.
That means it suits to the market price movement.
Trader can use it to predict the price movement.
Business man utilize it for decision.

How Chebyshev's Inequality Help Us in The Market

First, we need to know what we can achieve when we count out the result of this formula. The formula (1) tells us, the probability up bound is surely when you set the k value.
12 trader have the same $5,000 initiate fund. one month later, the average of their funds is $8,000, and the deviation is $4,500.

<br /> \begin{equation}<br /> P(|17000-8000|>=2*4500) \leq \frac{1}{2^2} \tag{example}<br /> \end{equation}<br />
We may know that at least 3(12 * 1/4) traders double their accounts.
Second, give a real example, the one we all familiar with, Bollinger Bands.