Random variables
Data arise from observations on variables that are measured on different scales. A nominal scale is used for named categories (eg race, gender) and an ordinal scale for data that can be ranked (eg attitudes, position) - no arithmetic operations are valid with either. Interval scale measurements can be added and subtracted only (eg temperature), but with ratio scale measurements (eg age, weight) multiplication and division can be used meaningfully as well. Generally, random variables are either discrete or continuous.
Note: all data are discrete because the accuracy of measuring is always limited.A discrete random variable $$X$$ can take one of the values $$x_1, x_2, \ldots$$, the probabilities $$p_i = P(X=x_i)$$ must satisfy $$p_i\geq 0$$ and $$p_1 + p_2 + \ldots = 1$$. The pairs $$(x_i , p_i)$$ form the probability mass function (pmf) of $$X$$.
A continuous random variable $$X$$ takes values from a continuous set of possible values. It has a probability density function (pdf) $$f(x)$$ that satisfies $$f(x)\geq 0$$ and $$\displaystyle\int f(x)\,dx= 1$$, with $$\displaystyle P(a < X \leq b)=\int_a^b f(x) \, dx$$.
Cumulative Distribution FunctionThis is defined as a function of any real value $$t$$ by \[F(t) = P(X\leq t).\] If $$X$$ is a continuous random variable if $$F$$ is a continuous function of $$t$$; if $$X$$ is discrete, then $$F$$ is a step function.