Random variables are an important concept of probability theory. It can be said as the very basic concept of probability theory. Random variables are those whose values are dependent on the outcomes of some random experiments. Random variables can be categorized as discrete and continuous. The probability distribution can be constructed for a given random variable. 

Discrete Random Variables

Discrete random variables are those who take discrete values, that is, countable values such as $0, 1, 2, 3,$ ... Some examples of discrete random variables are number of Kings coming while choosing n cards from deck of $52$ cards. The probability of discrete random variable is the probability related to each of these discrete values. For example, four cards are chosen from the deck of $52$ cards then find the probability of getting three red cards. Here, the number of red cards is the discrete random variable and the probability attached is the probability of getting $0, 1, 2, 3,$ or $4$ red cards out of four cards chosen.

Continuous Random Variable

A continuous random variable is one which can take up infinite random values. For example, height of students in a class can have infinite number of values and hence it will be called continuous random variable. There is no specific value for a continuous random variable and hence the it is defined over an interval of values. The probability of a continuous random variable $X$ can be defined as the area of a curve,

Constructing a Probability Distribution

A probability distribution is a pictorial representation of the probabilities of different values of random variable. For example, out of three bulbs the probability distribution of the random variable of number of defective bulbs can be given as,

 X  0  1  2  3
 P(X)  0.19  0.65 0.34

where the probabilities for different values of the random variable $X$ has already been given. In general, to construct a probability distribution for a random variable $X$ the probabilities for different values of $X$ should be calculated and written in tabular form or in a chart.

Functions of a Random Variable

Functions of a random variable can be obtained using different techniques. If $X$ is a random variable and $Y = f(X)$, that is, $Y$ is a function of $X$ then $Y$ is also a random variable. The probability distribution of $Y = f(X)$ can be found using the given techniques:

1) Distribution function technique

Change of variable technique

Two to one transformation

Probability Density Functions

Probability density function (pdf) is defined for a continuous random variable to find the relative probability of a given random variable $X$ to take up a certain value. It can also be defined as the equation used to define a continuous distribution. The properties of a probability density function for a continuous distribution are:

1) The graph of the probability density function is continuous over the range over which the random variable is defined.

2) The area bounded by the curve of the function equals one.

3) The probability between the values, say $a$ and $b$ of the random variable $X$ is the area of the curve bounded between $a$ and $b$.