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.

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

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

Median and Percentile

The a-th percentileis the point at which a percent the area under the curve is to one side. You want $P(X\leq x)$ to be a%, the calculation of which is in the page above.

By the same logic, the quartiles are at 25%, 50%, and 75% accordingly.

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

Uniform random variable is described by two parameters: $a$ is minimum, and $b$ is maximum. It has a rectangular distribution, where every point has the same probability density.

PDF

$f(x)={\begin{cases}{\frac {1}{b-a}}&a\leq x\leq b\\0&{\text{otherwise}}\end{cases}}$

CDF

$F(x)={\begin{cases}0&x<a\\{\frac {x-a}{b-a}}&a\leq x\leq b\\1&x>b\end{cases}}$

Mean

$\mu _{X}={\frac {a+b}{2}}$

Variance

$\sigma ^{2}={\frac {1}{12}}(b-a)^{2}$

Exponential Distribution $X\sim Exp(\lambda )$

The exponential distribution models events that occurs

• Continuously
• Independently
• At a constant average rate

Exponential distribution has the memoryless property: the probability to an event does not change no matter how much time has passed.

In probability terms, the probability that we must wait an additional $t$ units given that we have waited $s$ units

$P(T>t+s|T>s)=P(T>t)=e^{-\lambda t}$

Notably, it models time until some event has happened.

PDF

$f(x)={\begin{cases}\lambda e^{-\lambda x}&a\leq x\leq b\\0&{\text{otherwise}}\end{cases}}$

CDF

$F(x)=1-e^{-\lambda x}$

Mean

Integration by parts

$\mu _{X}={\frac {1}{\lambda }}$

Variance

Integration by parts

$\sigma ^{2}={\frac {1}{\lambda ^{2}}}$