Continuous Random Variables

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A continuous random variable maps outcomes to values of an uncountable set (e.g., the real numbers). For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values (such as an interval of non-zero length) may be positive.

Basic Definition

Cumulative distribution function for a continuous random variable X with a numerical value of interest a is $\underset{0}{\overset{a}{\int }}f(x)dx=F(a)$

Key Variables

${\mu }_X=E(X)$

${\sigma }_X^2=Var(X)=E[(X-{\mu }_X{)}^2]=E(X^2)-{\mu }_X^2$

Standardized Random Variable

$Z=\frac{X-{\mu }_X}{{\sigma }_X}$

$E(Z)=1,Var(Z)=1$

Normal Distribution

Probability Density Function

$f(x)=\frac{1}{\sqrt{2\pi {\sigma }_X^2}}{e}^{-\frac{(x-{\mu }_X{)}^2}{2{\sigma }_X^2}},\mathrm{\forall }x$