A random variable is a function which maps the events in the sample space of a random experiment to a real line. A random experiment may have several outcomes, the set of all those outcomes is known as sample space. There is no probability distribution for continuous random variables but there is probability distribution function and cumulative distribution function can be obtained from it on a given interval.

## Definition

Random variables can be continuous or discrete. The continuous random variable can take continuous values while the the discrete random variable takes the discrete values. The continuous values are the values that can be measured but not counted. If the probability density function of a continuous random variable $x$ is $f(x)$, then it can be written that for all $x,\ \int f(x)dx$ = $1$

The probability value of a continuous random variable is calculated in intervals.

## Probability Distribution for Random Variable

The probability distribution of a random variable is the table or list of probabilities attached with each value of the random variable. The probability distribution can be given for discrete random variables, that is, for each discrete value of the random variable the probability of its occurrence is given. Suppose, we roll a die then the discrete values for the random variable will be $\{1,\ 2,\ 3,\ 4,\ 5,\ 6\}$. The probability distribution for the random variable $X$ representing the outcome of the rolling of die is given below,

x |
P(x) |

1) |
$\frac{1}{6}$ |

2) |
$\frac{1}{6}$ |

3) |
$\frac{1}{6}$ |

4) |
$\frac{1}{6}$ |

5) |
$\frac{1}{6}$ |

6) |
$\frac{1}{6}$ |

## Cumulative Probability Distribution for Continuous Random Variable

As the probability of continuous random variable is calculated in intervals, the cumulative probability distribution function of the random variable can be calculated in the given interval. If the probability distribution function for a continuous random variable $x$ is given to be $f(x)$ then the cumulative probability distribution function is given by,

$F(t)$= $P(X \leq t)$ = $\int_{a}^{t}f(x)dx$

By differentiating the cumulative distribution function, the probability denisty function of a distribution can be obtained.

## Types of Distribution for a Continuous Random Variable

The distribution of values for a continuous random variable can be classified in two ways: Uniform distribution and Normal distribution.

Uniform distribution: When in an interval, say $[a, b]$, the probability is equal for all values of the random variable between $a$ and $b$ then it is known as uniform distribution.

Normal distribution: The distribution which can be represented as a bell-shaped curve is known as standard normal distribution.