9 Properties Of Mean And Variance Of Random Variables

The meaning of random is uncertain. So, a random variable is the one whose value is unpredictable. A variable, whose possible values are the outcomes of a random experiment is a random variable. In this article, students will learn important properties of mean and variance of random variables with examples.

The mean or expected value or the expectation is called an average in statistics and probability. The expected value can be calculated if the probability distribution for a random variable is found. Mean of a random variable defines the location of a random variable whereas the variability of a random variable is given by the variance.

9 Important Properties of Mean and Variance of Random Variables

Property 1: E (X + Y) = E (X) + E (Y). (X and Y are random variables)

Property 2: E (X₁ + X₂ + … + X_n) = E (X₁) + E (X₂) + … + E (X_n) = Σ_iE (X_i).

Property 3: E (XY) = E (X) E (Y). Here, X and Y must be independent.

Property 4: E [aX] = a E [X] and E [X + a] = E [X] + a, where a is a constant

Property 5: For any random variable, X > 0, E(X) > 0.

Property 6: E(Y) ≥ E(X) if the random variables X and Y are such that Y ≥ X.

Property 7: The variance of a constant is 0.

Property 8: V [aX + b] = a² V(X), where a and b are constants, X is random variable.

Property 9: V (a₁X₁ + a₂ X₂ + … + a_nX_n) = a₁² V(X₁) + a₂² V(X₂) + … + a_n² V(X_n).

Example on Variance of Random Variable

1] A fair six-sided die can be modeled as a discrete random variable, X, with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is

Solution:

\(\begin{array}{l}\mu_x = \frac{(1+2+3+4+5+6)}{6} = \frac{7}{2}\end{array} \)

px(x) = 1/6

Therefore, the variance of X is

Var(X) = E(X – μ_x)²

= ∑(x – μ_x)²px(x)

\(\begin{array}{l}{Var} (X)=\sum _{i=1}^{6}{\frac {1}{6}}\left(i-{\frac {7}{2}}\right)^{2}\\[5pt]\\={\frac {1}{6}}\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\right)\\[5pt]\\={\frac {35}{12}}\approx 2.92.\end{array} \)

The general formula for the variance of the outcome, X, of an n-sided die is

\(\begin{array}{l}{Var} (X)= {E} (X^{2})- {E} (X)^{2}\\[5pt]\\={\frac {1}{n}}\sum _{i=1}^{n}i^{2}-\left({\frac {1}{n}}\sum _{i=1}^{n}i\right)^{2}\\[5pt]={\frac {(n+1)(2n+1)}{6}}-\left({\frac {n+1}{2}}\right)^{2}\\[4pt]={\frac {n^{2}-1}{12}}\end{array} \)

Also, students are advised to solve JEE Past year solved Problems on Statistics and join BYJU’S online classes to crack the JEE Exam.

Video Lessons

Statistics – Important Topics

Mean, Median and Mode

Frequently Asked Questions

Define Variance.

Variance is the expected value of the squared variation of a random variable from its mean value.

Give the equation to find the Variance.

Variance is given by σ² = ∑(xi-x̄)²/N. x₁, x₂, …., x_N are the N observations.

Give a property of Variance.

The variance of a constant is 0.

Properties Of Mean And Variance Of Random Variables

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