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Cumulative Distribution Function 10/5

Definition:
Let X be a continuous real-valued random variable, its cumulative distribution function is:

\begin{displaymath}F_X(x)=P(X\leq x)\end{displaymath}

Theorem:
If X has a density f(x) then:

\begin{displaymath}1. \qquad F(x)= \int_{-\infty}^x f(t) dt\end{displaymath}

is the cdf and

\begin{displaymath}2. \qquad \frac{d}{dx} F(x)=f(x)\end{displaymath}

Proof:
Property 1. comes from the definition of probability as a function of the density:

\begin{displaymath}P(\left (-\infty, x \right ])= \int_{-\infty}^x f(t) \end{displaymath}

.
Property 2 is due to the fundamental theorem of calculus.



Susan Holmes
1998-12-07