Question

What is the formula for the variance of a probability distribution?

Answer 1

Hint: We know the variance of a random variable X is the mean or expected value of the squared deviation from the mean of X. Using this definition, we can arrive at a simpler formula for variance, both for continuous and discrete random variables.

Complete step-by-step answer:
We know that the variance measures how far a set of numbers is spread out, from their average value. Also, we are aware that variance is the square of standard deviation. The variance of a random variable X is represented by ${\sigma }^2,s^2$ or Var(X).
Using the definition of variance, for a random variable X, we can write the variance as
$Var\left(X\right)=E\left[{(X-E\left[X\right])}^2\right]$
where, E(X) represents the expected value or mean for the random variable X.
We can expand the above equation as
$Var\left(X\right)=E[X^2-2X\cdot E\left[X\right]+E{\left[X\right]}^2]$
Hence, we can write
$Var\left(X\right)=E\left[X^2\right]-2E\left[X\right]\cdot E\left[X\right]+E{\left[X\right]}^2$
Thus, we can simplify this to get
$Var\left(X\right)=E\left[X^2\right]-E{\left[X\right]}^2...\left(i\right)$
Hence, we can also say that the variance of X is equal to the difference of the mean of the square of X and the square of the mean of X.
We know that for a continuous random variable X, the total probability is
$P(a\le X\le b)=\underset{a}{\overset{b}{\int }}f\left(x\right)dx=1$.
Also, we can write the expected value as,
$E\left(X\right)=\underset{a}{\overset{b}{\int }}xf\left(x\right)dx$.
Hence, we can write the variance using equation (i) as
$Var\left(X\right)=\underset{a}{\overset{b}{\int }}{x}^{2}f\left(x\right)dx-{\left\{\underset{a}{\overset{b}{\int }}xf\left(x\right)dx\right\}}^2$.
Similarly, for a discrete random variable X, we have the total probability as,
$\sum _{i=1}^{n}P\left(x_i\right)=1$.
Also, we can write the expectation of X as,
$E\left(X\right)=\sum _{i=1}^n{x}_{i}P\left(x_i\right)$.
Thus, we can write the variance using equation (i) as
$Var\left(X\right)=\sum _{i=1}^n{x_i}^{2}P\left(x_i\right)-{\left\{\sum _{i=1}^n{x}_{i}P\left(x_i\right)\right\}}^2$.

Note: We must take care that in the above explanation, we have assumed that the probability distribution of continuous random variable to be defined from a to b, and the discrete random variable is assumed to have n different possibilities.