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Concept of Distribution

A numerical outcome $ X$ from an experiment is called a random variable (r.v.). We conventionally denote random variables by uppercase letters $ X, Y, Z, U, V, \ldots$ from the end of the alphabet. A discrete random variable is a random variable taking its value on a finite set $ \{a_1,a_2,\ldots,a_n\}$ of real numbers (usually integers), or on a countably infinite set $ \{a_1,a_2,a_3,\ldots\}$. The statement such as ``$ X = a_i$'' is an event, and the probability of the event $ \{X = a_i\}$ is denoted by $ P(X = a_i)$. The function

$\displaystyle p(a_i) := P(X = a_i)
$

over the possible values of $ X$, say $ a_1,a_2,\ldots$, is called a frequency function, or a probability mass function.

Continuous random variable. A continuous random variable is a random variable whose possible values are real values such as 78.6, 5.7, 10.24, and so on. Examples of continuous random variables include temperature, height, diameter of metal cylinder, etc. In what follows, a random variable always means a ``continuous'' random variable, unless it is particularly said to be discrete. The probability distribution of a random variable $ X$ specifies how its values are distributed over the real numbers. This is completely characterized by the cumulative distribution function (cdf). The cdf

$\displaystyle F(t) := P(X \le t).
$

represents the probability that the random variable $ X$ is less than or equal to $ t$. Then we often say that ``the random variable $ X$ is distributed as $ F$.''

Probability distributions. It is often the case that the probability of the random variable $ X$ being in any particular range of values is given by the area under a curve over that range of values. This curve is called the probability density function (pdf) of the random variable $ X$, denoted by $ f(x)$. Thus, the probability that `` $ a \le X \le b$'' can be expressed as

$\displaystyle P(a \le X \le b) = \int_a^b f(x) dx .
$

Furthermore, we can find the following relationship between cdf and pdf:

$\displaystyle F(t) = P(X \le t) = \int_{-\infty}^t f(x) dx.
$


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