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Introduction
When a statistical phenomenon or experiment is described in numerical terms, the result is a random variable. A random variable can be classified into discrete or continuous categories. Random variables are associated with probability distributions. These are the types of distributions representing all possible outcomes or values presented by a random variable from a given range of values.
Probability distribution of random variables
For a discrete random variable, the probability curve is defined by the function of the random variable. This function gives the corresponding value of the probability for each outcome.
Discrete random variable
A discrete random variable represents a finite sequence or number. In other words, a discrete random variable represents a countable set of observations or values. It mainly consists of individual values. Examples of discrete random variables include the number of cycles sold on a particular day at a store, the spanner inventory of a hardware store, and the number of students in a class.
For a discrete random variable, the probability distribution lists the individual probabilities associated with all the possible individual outcomes of the random variable. The probabilities associated with the outcomes must satisfy certain conditions as follows:
- The probability of each value must be more than or equal to zero and less than or equal to one. This indicates that the probability should be a fraction.
- The sum of all the probabilities should be equal to 1.
Let us assume that a random variable T can take the values 2, 3, 5, and 7. The probabilities associated with each outcome are given in the following table:
The probability that the random variable is equal to two of the outcomes is the sum of the probability of the two probabilities. Thus, the probability that T is 3 or 7 is 0.3 + 0.4 = 0.7
and the probability that T is greater than 2 is 1 – 0.1 = 0.9
Continuous random variable
A random variable that can take on any number of values on the real number line is a continuous random variable. Examples of continuous random variables include height, weight, and time duration of running some distance. Countable terms or values do not represent a continuous random variable. Instead, it is present within a range of values represented by a curve. The area under the curve represents the possible outcomes of the continuous random variable. Hence, this variable type does not have any specific set of individual values. A noteworthy characteristic of continuous random variables is that the possibility of the variable having any individual value is zero. This is because the outcomes are not individual but are a part of a continuous curve and the number of values is infinite.
For a continuous random variable Y, the probability that it will be in the set of outcomes B is defined by the area above the value of B under the curve defined by a function. Thus, the function defining the range must satisfy the following conditions:
- The total area under the curve should be equal to 1.
- The curve should not have any negative value.
Uniform distribution
A uniform distribution, also known as a rectangular distribution, has a constant probability. The probability is uniform because the values in such a distribution have equal opportunities for observation, indicating that the curve has a constant height for the duration of the interval and not height elsewhere.
Conclusion
A random variable is an entity used most commonly in statistics and probability. It is used for quantifying all the possible outcomes that may result from a random phenomenon. The values of the random variable are usually defined by a function, which assigns a probability to each possible outcome. Econometric or regression analysis makes use of random variables. Any student of mathematics or statistics or even economics must have reliable random variable study material to properly understand the subject. Study material notes on random variables are imperative for a well-rounded mathematics and statistics study.
