Continuous Random Variable

Continuous random variables have an inifinite number of values for any given interval. While similar, the approach to analysis is very different from discrete variables

• Summation becomes integration
• Probability becomes area under a curve

Probability Distribution Function $f(x)$

The probability density function (pdf) maps a continuous variable to a probability density.

As the name "density" suggests, the area under the pdf curve between a range is the probability of the variable being in that range.

$P(c\leq x\leq d)=\int _{c}^{d}f(x)dx=F(d)-F(c)$

Total area under the curve must be $1$, as chances of events happening is 100% if the range includes all possible events.

$\int _{-\infty }^{\infty }f(x)dx=1$

There is no area under a single point

$P(X=a)=0$

Mean and Variance

The mean and variance calculations are pretty much the same as that of discrete random variables, except the summations are swapped out for integrals.

$E(X)=\mu _{X}=\int _{-\infty }^{\infty }xf(x)dx$

$Var(X)=\sigma _{X}^{2}=\int _{-\infty }^{\infty }(x-\mu _{X})^{2}f(x)dx=\int _{-\infty }^{\infty }x^{2}f(x)dx-\mu _{X}^{2}$

Uniform Distribution $X\sim Uniform(a,b)$

$a$ is minimum, and $b$