Introduction

Introduction#

A probability density function (PDF) defines the continuous probability distribution. The PDF follows two basic rules,

  1. \[ \int_{-\infty}^{\infty} f(x) ~dx = 1 \]
  2. \[ P(a < X \le b) = \int_a^b f(x)~ dx \]

Where \(f(x)\) is the conventional syntax for PDF.

Formalism#

To deal with the existential crisis that the chance of a single-value event \(\set{X=x}\) is,

\[ P(X = x) = 0 \]

because the area for at \(X=x\) is zero (rather poorly defined), we develop the Reynmann interpretation of integrals to the formalism of continuous probability. Take \(dx\) as some small infinitestimal width for some interval \((x-dx, x+dx)\) and define \(dx\) to be an event such that,

\[ P(X \in dx) \equiv P(x-dx < X \le x+dx) = \int_{x - dx}^{x+dx} f(x) dx \]

Asymptotically, as \(dx \to 0\) the integral dissapears to become the Reinman integral,

\[ \boxed{P(X \in dx) \sim f(x) dx} \]

A physics formality defines the differential form of the probability ,

\[ dP = fdx \]

Cumulative Distribution Function#

The cumulative distribution function (CDF) is defined the left limit to negative infinity,

\[ F(x) = P(X \le b) = \int_{-\infty}^b f(x)~ dx \]

Where \(F(x)\) is the conventional syntax for CDF.

Relation to PDF : By the fundamental theorem of calculus,

$$
\begin{gather*}
P(a < X \le b) = \int_a^b f(x)~ dx = F(b) - F(a)\\
\Big\Downarrow\\
f = \frac{dF}{dx}
\end{gather*}
$$

Expectation#

Analgous to discrete case, the expectation is,

\[ E(g(X)) = \int_{-\infty}^\infty g(x)f(x)~dx \]

Existence : The expectation exists only if the integral exists. A useful theorem is if,

$$
E(g(X)) = \int_{-\infty}^\infty \abs{g(x)}f(x)~dx < \infty
$$

then the expectation exists.
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