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 $\int \limits _{0}^{a}f(x)dx=F(a)$

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

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

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

Normal Distribution

Probability Density Function

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