Chapitre 4

The set of potential outcomes of a random experiment (a family of random variables indexed by time t) defines a random variable X [9]. It is characterised by a distribution function Fx, which is the probability that X is less than or equal to a real number x (time fixed at t=ti , thus having a random variable ‘RV') :

\(F_x=\text{prob}(X \leq x)\) (1)

• Probability Density

The probability density of a random variable X is, by definition, the derivative of the distribution function.

\(f_ {X} ({x})= \frac{ dF_ {X}({x})}{d {x} }\) (2)

The integral of the probability density from -∞ to x1 gives the cumulative distribution function :

\({F}_{X}=\int_{-\infty}^{{x}_{1}}{{f}_{X}({x}){dx}}\) (3)

The probability density is always a normalised function :

\(\int_{-\infty}^{+\infty}{{f}_{X}{dx}}={1}\) (4)

• Expectation

  \({E}\left({X}\left({t}\right)\right)=\int_{-\infty}^{+\infty}{{x}{f}_{X}}\left({x}\right){dx}\) (5)

• Average, or moment of order 1

  \({m}_{{X}_f{1}}({t})={E}\left({X}_{1}\right)=\int_{-\infty}^{+\infty}{{X}_f{1}{f}_{X}}\left({x}\right){dx}\) (6)

•  The moment of order K is defined by :

  \({m}_{X}({t})={E}\left({X}^{k}\right)=\int_{-\infty}^{+\infty}{{X}^{k}{f}_{X}}\left({x}\right){dx}\) (7)

• The variance of X is defined by :

  \({\sigma}_{x}^f{2}\left({t}\right)={Var}\left({X}\right)={E}\left[{X}-{{E}\left({X}\right)}^{2}\right]={E}\left[{X}^{2}\right]-{({E}\left[{X}\right])}^\mathbf{2}\) (8)