Variance Of Random Variables In Probability And Statistics

Variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by o², s², Var(X), V(X), or V(X).

Random variable

A random variable (also called the d random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events.

Definition of Random variable

A random variable is a measurable function from a probability measure space (called the sample space) to a measurable space. This allows consideration of the pushforward measure, which is called the distribution of the random variable; the distribution is thus a probability measure on the set of all possible values of the random variable. Two random variables may have identical distributions but differ significantly; for instance, they may be independent.

A random variable X is a measurable

Function X: →E from a set of possible outcomes to a measurable space E. The axiomatic technical definition requires two as a sample space of a probability triple (N, F, P) (see the measure-theoretic definition). 

Example –

If a random variable X: → R defined on the probability space (N, F, P) is given, we can ask questions like “How likely is it that the value of X is equal to 2?”. This is the same as the probability of the event {w: X(w) = 2} which is often written as P(X= 2) or px (2) for short.

Recording all these probabilities of output ranges of a real-valued random variable X yields the probability distribution of X. The probability distribution “forgets” about the particular probability space used to define X and only records the probabilities of various values of X. Such a probability distribution can always be captured by its cumulative distribution function.

 Fx(x) = P(X ≤ x)

Random variables are of three types. 

1. Discrete random variable 

2. Continuous random variable

3. Mixed type of random variable

Discrete random variable

When X’s image (or range) is countable, the random variable is called a discrete random variable. Its distribution is a discrete probability distribution, i.e., it can be described by a probability mass function that assigns a probability to each value in the image of X. 

Continuous random variable

A continuous random variable is a random variable whose cumulative distribution function is continuous everywhere. There are no gaps, which correspond to numbers with a finite probability of occurring.

It is denoted by 

Vc ER: Pr(X= c) = 0

Mixed random variables

A mixed random variable is a random variable whose cumulative distribution function is neither discrete nor everywhere-continuous.

Example, 

If a coin is flipped and the spinner is spun only if the result of the coin toss is headed. If the result is tails, X = −1; otherwise, X = the value of the spinner as in the preceding example. There is a probability of 1⁄2 that this random variable will have the value. 

In this case, the observation space is the set of real numbers. Recall, (N, F, P) is the probability space. For a real observation space, the function X:→ Ris a real-valued random variable if

{w: X(w) ≤r} EF

Vr ER.

Real valued random variables

This definition is a special case of the above because of the set. 

{(-∞,r]: r ≤ R} generates the Borel o-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that,

{w: X(w) ≤r} = X–¹ ((-∞, r]). 

The probability distribution of a random variable is often characterized by a small number of parameters, which also have a practical interpretation. 

For example, it is often enough to know its “average value.” This is captured by the mathematical concept of the expected value of a random variable, denoted E[X], called the first moment.

In general, E[ƒ(X)] is not equal to f(E[X]). Once the “average value” is known, one could then ask how far from this average value the values of X typically are, answered by the variance and standard deviation of a random variable.

E[X] can be viewed intuitively as an average obtained from an infinite population, the members of which are individual evaluations of X.

Mathematically, this is known as the (generalized) problem of moments: for a given class of random variables X, find a collection {f} of functions such that the expectation values E[f; (X)] fully characterize the distribution of the random variable.

Conclusion

By learning this concept, we can calculate the values of random variables by different methods. The random variables in math can give the measure of spread for a random variable distribution that determines the degree to which the values of a random variable differ from the expected value. It’s the measure of dispersion the most often used, along with the standard deviation, which is simply the square root of the variance. The advantage of variance is that it treats all deviations from the mean as the same regardless of their direction. Using the random variables, we can find the expected values of it.