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

Concept

• 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

$$F(x) = P(X \leq x) \quad \forall x \in R$$

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

$$f(x) = \frac{dF(x)}{dx}$$

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

$$\mu = E(X) = \int_{-\infty}^{\infty} xf(x) dx$$ $$\sigma^2 = Var(X) = E[X^2] - E[X]^2 = \int_{-\infty}^{\infty} x^2 f(x)dx - \mu^2$$

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

$$E(g(x)) = \int_{-\infty}^{\infty} g(x)f(x) dx$$

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

$$\iff P(X = x_i, Y = y_j) = P(X = x_i) \cdot P(Y = y_j) \quad \forall x_i, y_j$$

Example

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.

$$P(Y \leq 1) = \frac{1}{4}$$ $$P(Y \leq 2) = \frac{1}{2}$$ $$P(3 < Y \leq 3.5) = \frac{1}{8}$$

In the example above

$$F(y) = 0 \; (y < 0), \quad F(y) = \frac{y}{4} \; (0 \leq y < 4), \quad F(y) = 1 \; (4 \leq y)$$ $$f(y) = F'(y) = \frac{1}{4} \; (0 \leq y < 4)$$ $$\mu = \int_{0}^{4} y \cdot \frac{1}{4} = [\frac{1}{8}y^2]_0^4 = 2$$ $$\sigma^2 = \int_{0}^{4} \cdot y^2 \frac{1}{4} - 2^2= [\frac{1}{12}y^3]_0^4 - 4 = \frac{4}{3}$$

Next

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.