illustrative abstractions


Continuous Random Variables VS. Discrete Random Variables (Part 1)


  • Discrete random variable : countable number of possible values
  • Continuous random variable : uncountably (infinite) number of possible values

  • The cumulative distribution function of a variable $X$ is the function

  • The probability density function of a variable $X$ is the function

A given point on $f(x)$ should not be interpreted as $f(x) = P(X = x)$. It should be an analogy to density, or change of probability rate at $x$.

Properties for $f(x)$,

  • $f(x) \geq 0, \forall x \in R $

  • $\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a) = P(a < X \leq B) $

  • $\int_{-\infty}^{\infty}f(x) dx = P(-\infty < X \leq + \infty) = 1 $

Population Mean

For any transformation made on $x$, let $g(x)$

Two continuous random variables $X$ and $Y$ are independant


Given a unit circle marked with $0, 1, 2, 3$ at $\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ respectively. The circumference of the unit circle has value range from $0 \leq Y < 4$.

A pointer from the origin with length of $1$ is spun and is equally likely to stop anywhere on the circle. Let $Y$ be the value obtained from the circle.

$Y$ is a continuous random variable.

In the example above


In the next post, we will be looking at continuous distributions, as similar to discrete distributions in previous posts here, part 1 and here, part 2.