Discrete random variables are those that can take on a countable number of distinct values.
The variance of a discrete random variable X, denoted as Var(X), measures the spread of the values around the mean. It is defined as:
$$ Var(X) = E[(X - \mu)^2] $$
where $\mu = E(X)$ is the expected value or mean of X.
An alternative and often more convenient formula for calculating variance is:
$$ Var(X) = E(X^2) - [E(X)]^2 $$
This formula is particularly useful in practice as it often simplifies calculations.
Continuous random variables can take on any value within a given range. They are described by probability density functions (PDFs). In other words, for a continuous random variable, $P(a \le X \le b)=\int_a^b f(x)\,dx$ and $P(X=x)=0$.
A probability density function f(x) for a continuous random variable X has the following properties:
Unlike discrete probability mass functions, f(x) can take values greater than 1, as long as the total area under the curve is 1.
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