Difference Between Random Variables and Probability Distribution

Random Variables vs Probability Distribution

Statistical experiments are random experiments that can be repeated indefinitely with a known set of outcomes. Both random variables and probability distributions are associated with such experiments. For each random variable, there is an associated probability distribution defined by a function called cumulative distribution function.

What is a random variable?

A random variable is a function that assigns numerical values to the outcomes of a statistical experiment. In other words, it is a function defined from the sample space of a statistical experiment into the set of real numbers.

For example, consider a random experiment of flipping a coin twice. The possible outcomes are HH, HT, TH and TT (H – heads, T – tales). Let the variable X be the number of heads observed in the experiment. Then, X can take the values 0, 1 or 2, and it is a random variable. Here, the random variable X will map the set S = {HH, HT, TH, TT} (the sample space) to the set {0, 1, 2} in such a way that HH is mapped to 2, HT and TH are mapped to 1 and TT is mapped to 0. In function notation, this can be written as, X: S → R where X(HH)=2, X(HT)=1, X(TH)=1 and X(TT)=0.

There are two types of random variables: discrete and continuous, accordingly the number of possible values a random variable can assume is at most countable or not. In the previous example, the random variable X is a discrete random variable since {0, 1, 2} is a finite set. Now, consider the statistical experiment of finding the weights of students in a class. Let Y be the random variable defined as the weight of a student. Y can take any real value within a specific interval. Hence, Y is a continuous random variable.

What is a probability distribution?

Probability distribution is a function that describes the probability of a random variable taking certain values.

A function called cumulative distribution function (F) can be defined from the set of real numbers to the set of real numbers as F(x) = P(X ≤ x) (the probability of X being less than or equal to x) for each possible outcome x. Now the cumulative distribution function of X in the first example can be written as F(a)=0, if a<0; F(a)=0.25, if 0≤a<1; F(a)=0.75, if 1≤a<2 and F(a)=1, if a≥2.

In case of discrete random variables, a function can be defined from the set of possible outcomes to the set of real numbers in such a way that ƒ(x) = P(X = x) (the probability of X being equal to x) for each possible outcome x. This particular function ƒ is called the probability mass function of the random variable X. Now the probability mass function of X in the first particular example can be written as ƒ(0)=0.25, ƒ(1)=0.5, ƒ(2)=0.25, and ƒ(x)=0 otherwise. Thus, probability mass function along with the cumulative distribution function will describe the probability distribution of X in the first example.

In the case of continuous random variables, a function called the probability density function (ƒ) can be defined as ƒ(x) = dF(x)/dx for each x where F is the cumulative distribution function of the continuous random variable. It is easy to see that this function satisfies ∫ƒ(x)dx = 1. The probability density function along with the cumulative distribution function describes the probability distribution of a continuous random variable. For example, the normal distribution (which is a continuous probability distribution) is described using the probability density function ƒ(x) = 1/√(2πσ²) e^([(x-µ)]²/(2σ²)).