Probability – - Alternative Formula For Evaluation Of Variance

Probability - Alternative formula for evaluation of Variance

In probability theory, variance is a measure of how much the values in a data set deviate from the mean.
It measures the spread or dispersion of the data.
The formula to calculate the variance of a data set is given by:
- Var(X) = Σ(x - μ)² * P(x), where x represents each value in the data set, μ is the mean of the data set, and P(x) is the probability of x occurring.
However, in some cases, it is not easy to calculate the variance using the above formula.
In such cases, an alternative formula can be used, which simplifies the calculation.

Alternative Formula for Variance Calculation

The alternative formula for calculating the variance is given by:
- Var(X) = E(X²) - [E(X)]², where E(X) represents the expected value of X and E(X²) represents the expected value of X squared.
This formula is particularly useful when working with discrete random variables.
It involves calculating the expected value of X and the expected value of X squared.
Let’s understand this formula with the help of an example.

Example 1

Suppose we have a fair six-sided dice.
Let X be the random variable representing the outcome of rolling the dice.
The possible values of X are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6.
We want to calculate the variance of X using the alternative formula.
Step 1: Calculate E(X)
- E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5
Step 2: Calculate E(X²)
- E(X²) = (1² * 1/6) + (2² * 1/6) + (3² * 1/6) + (4² * 1/6) + (5² * 1/6) + (6² * 1/6) = (1 + 4 + 9 + 16 + 25 + 36) / 6 = 15.1667
Step 3: Calculate Var(X)
- Var(X) = E(X²) - [E(X)]² = 15.1667 - (3.5)² = 15.1667 - 12.25 = 2.9167

Properties of Variance

Variance has the following properties:
- Var(X + c) = Var(X), where c is a constant
- Var(cX) = c²Var(X), where c is a constant
- Var(X + Y) = Var(X) + Var(Y), where X and Y are independent random variables
- Var(aX + bY) = a²Var(X) + b²Var(Y) + 2abCov(X, Y), where a and b are constants and Cov(X, Y) represents the covariance between X and Y.
These properties can be used to simplify the calculation of variance in various scenarios.

Example 2

Consider two independent random variables X and Y.
The variance of X is 4 and the variance of Y is 9.
Let Z = 2X - 3Y.
We want to calculate the variance of Z using the properties of variance.
Step 1: Calculate Var(Z)
- Var(Z) = (2²)Var(X) + (-3)²Var(Y) + 2(2)(-3)Cov(X, Y) = 4(4) + 9(9) + 2(2)(-3)(0) = 16 + 81 + 0 = 97
Therefore, the variance of Z is 97.

Conclusion

In probability theory, variance is a measure of the spread or dispersion of a data set.
The alternative formula for calculating variance, Var(X) = E(X²) - [E(X)]², is useful when the traditional formula is not easily applicable.
Variance has certain properties that can simplify its calculation in different scenarios. Here are slides 11 to 20 for the topic “Probability - Alternative formula for evaluation of Variance”:

Slide 11:

Probability - Alternative formula for evaluation of Variance

The alternative formula for calculating the variance is given by:
- Var(X) = E(X²) - [E(X)]²

Slide 12:

Example 1: Rolling a Fair Six-Sided Dice

Suppose we have a fair six-sided dice.
Let X be the random variable representing the outcome of rolling the dice.
The possible values of X are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6.
We want to calculate the variance of X using the alternative formula.

Slide 13:

Example 1: Rolling a Fair Six-Sided Dice (contd.)

Step 1: Calculate E(X)
- E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5

Slide 14:

Example 1: Rolling a Fair Six-Sided Dice (contd.)

Step 2: Calculate E(X²)
- E(X²) = (1² * 1/6) + (2² * 1/6) + (3² * 1/6) + (4² * 1/6) + (5² * 1/6) + (6² * 1/6) = (1 + 4 + 9 + 16 + 25 + 36) / 6 = 15.1667

Slide 15:

Example 1: Rolling a Fair Six-Sided Dice (contd.)

Step 3: Calculate Var(X)
- Var(X) = E(X²) - [E(X)]² = 15.1667 - (3.5)² = 15.1667 - 12.25 = 2.9167

Slide 16:

Properties of Variance

Variance has the following properties:
- Var(X + c) = Var(X), where c is a constant
- Var(cX) = c²Var(X), where c is a constant
- Var(X + Y) = Var(X) + Var(Y), where X and Y are independent random variables
- Var(aX + bY) = a²Var(X) + b²Var(Y) + 2abCov(X, Y), where a and b are constants and Cov(X, Y) represents the covariance between X and Y.

Slide 17:

Example 2: Independent Random Variables

Consider two independent random variables X and Y.
The variance of X is 4 and the variance of Y is 9.
Let Z = 2X - 3Y.
We want to calculate the variance of Z using the properties of variance.

Slide 18:

Example 2: Independent Random Variables (contd.)

Step 1: Calculate Var(Z)
- Var(Z) = (2²)Var(X) + (-3)²Var(Y) + 2(2)(-3)Cov(X, Y) = 4(4) + 9(9) + 2(2)(-3)(0) = 16 + 81 + 0 = 97

Slide 19:

Example 2: Independent Random Variables (contd.)

Therefore, the variance of Z is 97.

Slide 20:

Conclusion

In probability theory, variance measures the spread or dispersion of a data set.
The alternative formula for variance, Var(X) = E(X²) - [E(X)]², provides a simplified calculation.
Variance has properties that can simplify its calculation in different scenarios.

Variance of a Continuous Random Variable

In addition to discrete random variables, variance can also be calculated for continuous random variables.
The formula for calculating the variance of a continuous random variable is slightly different from the formula for discrete random variables.
For a continuous random variable X with probability density function f(x), the variance is given by:
- Var(X) = ∫[(x - μ)² * f(x)] dx, where μ is the mean of the random variable.

Example 1: Calculating the Variance of a Continuous Random Variable

Suppose we have a continuous random variable X with the probability density function f(x) = 2x for 0 ≤ x ≤ 1.
We want to calculate the variance of X.
Step 1: Calculate the mean, μ.
- μ = ∫[x * f(x)] dx = ∫[x * 2x] dx = 2∫[x²] dx = 2[x³/3] from 0 to 1 = 2/3

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X).
- Var(X) = ∫[(x - μ)² * f(x)] dx = ∫[(x - 2/3)² * 2x] dx = 2∫[(x - 2/3)² * x] dx

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X) (contd.)
- We solve this integral using integration by parts.
  - Integration by parts formula: ∫[u dv] = uv - ∫[v du]
- Let u = (x - 2/3)²
  - du/dx = 2(x - 2/3)
- Let dv = x dx
  - v = x²/2

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X) (contd.)
- Using the integration by parts formula, we have:
  - ∫[(x - 2/3)² * x] dx = [(x - 2/3)² * (x²/2)] - ∫[(x²/2) * 2(x - 2/3)] dx

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X) (contd.)
- Using the integration by parts formula, we have:
  - ∫[(x - 2/3)² * x] dx = [(x - 2/3)² * (x²/2)] - ∫[(x²/2) * 2(x - 2/3)] dx

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X) (contd.)
- Simplifying the equation, we have:
  - [(x - 2/3)² * (x²/2)] - ∫[(x²/2) * 2(x - 2/3)] dx = (x - 2/3)³/2 - [x³/3 - 4x²/3 + 8x/9] from 0 to 1

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X) (contd.)
- Substituting the limits of integration, we have:
  - (1 - 2/3)³/2 - [(1)³/3 - 4(1)²/3 + 8(1)/9] - (0 - 2/3)³/2 + [(0)³/3 - 4(0)²/3 + 8(0)/9]

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X) (contd.)
- Simplifying the expression, we have:
  - (1/3)³/2 - (1/3 - 4/3 + 8/9) - (-2/3)³/2 + (0 - 0 + 0)

Example 1: Calculating the Variance of a Continuous Random Variable (contd.)

Step 2: Calculate the variance, Var(X) (contd.)
- Simplifying the expression, we have:
  - (1/27) - (3/9) - (-8/27) = 1/27 - 1/3 + 8/27 = 8/27 - 1/3 = 8/27 - 9/27 = -1/27