A random variable is also known as a stochastic variable and is a variable whose value is unknown. It can also be defined as a function that associates a special numerical value with each outcome of an experiment e.g. tossing a coin may either give a head or a tail, here denoting X to represent all heads is what a random variable is. Other examples that will give you an easy understanding continuous variable include: throwing two dice; X=sum of the numbers facing up i.e. among the numbers 1-6, throw a dice over and over until you get a six; X= the number of throws. Also without laying much stress on the die, we can have a more practical one i.e. the number of goals scored by a soccer team player X= the number of goals scored by the player in the season, here X maybe 0,1,2,3…
There are two types of random variables namely: discrete and continuous. A discrete random variable has a set of countable numbers (integers) with each value in the range having a probability of either zero or greater than zero. Example of a discrete random variable is taking the number of dollars in a randomly chosen bank account. Discrete random variable is further divided to either finite or infinite random variable. Finite discreet can only take known many values like the outcomes of rolling a die. Infinite discreet can take unlimited number of values like the count of the stars in the whole universe.
Continuous random variable contains a set of values that are completely uncountable. It can take any values within the range; a good example is the measure of your height or your current temperature.
Probability distribution of a random variable:
A probability distribution works hand in hand with the random variable interval either discrete or continuous to determine the likely hood of an event falling within a particular interval. In case of a continuous variable the probability distribution will be used to describe the range of all possible values that the random variable can get and the probability of the value of a measurable subset in the same range.
Probability distribution is used because the use of simple numbers to describe a quantity may turn out to be inadequate. For instance, the spread of a variable in almost any measurements i.e. durability of metals, traffic flow, people’s height or even sales growth will definitely apply some aspect of probability distribution. In physics many studies are linked to probability distribution like the kinetic properties of gases to the intense quantum mechanics of fundamental particles.
Various probability distributions are used in different applications. The most important and common ones that are frequently used include the Gaussian/normal distribution and the categorical distribution. The normal distribution has properties like taking a bell-shaped curve and it is used to approximate natural occurring distribution over real numbers. The categorical distribution on the other hand is used to give descriptions of experiments with finite and fixed numbers of outcomes. An example of a categorical distribution may be attributes to tossing a coin where possible outcomes are either a head or a tail with each having an equal chance of 0.5.