Calculating Variance of a Random Variable from Probability Distribution

To find the variance of a random variable \( X \) given its probability distribution, we follow a standard procedure involving calculating the expected value of \( X \) and the expected value of \( X^2 \).

The variance, denoted as \( Var(X) \) or \( \sigma^2 \), measures how spread out the values of the random variable are from its expected value (mean). The formula for variance is:

\( Var(X) = E(X^2) - [E(X)]^2 \)

where \( E(X) \) is the expected value (mean) of \( X \) and \( E(X^2) \) is the expected value of \( X^2 \).

Understanding the Probability Distribution Table

The probability distribution of the random variable \( X \) is provided in the table:

This table lists each possible value \( x_i \) that the random variable \( X \) can take and its corresponding probability \( P(X=x_i) \). Remember that the sum of all probabilities must equal 1: \( \frac{1}{3} + \frac{1}{6} + \frac{1}{6} + \frac{1}{3} = \frac{2}{6} + \frac{1}{6} + \frac{1}{6} + \frac{2}{6} = \frac{6}{6} = 1 \).

Calculating the Expected Value E(X)

The expected value \( E(X) \) is the weighted average of the possible values of \( X \), where the weights are the probabilities. The formula is:

\( E(X) = \sum x_i P(x_i) \)

Let's calculate \( E(X) \) using the given table values:

\( E(X) = (1 \times P(X=1)) + (3 \times P(X=3)) + (5 \times P(X=5)) + (7 \times P(X=7)) \)

\( E(X) = \left(1 \times \frac{1}{3}\right) + \left(3 \times \frac{1}{6}\right) + \left(5 \times \frac{1}{6}\right) + \left(7 \times \frac{1}{3}\right) \)

\( E(X) = \frac{1}{3} + \frac{3}{6} + \frac{5}{6} + \frac{7}{3} \)

Simplify the fractions:

\( E(X) = \frac{1}{3} + \frac{1}{2} + \frac{5}{6} + \frac{7}{3} \)

To add these, find a common denominator, which is 6:

\( E(X) = \frac{1 \times 2}{3 \times 2} + \frac{1 \times 3}{2 \times 3} + \frac{5}{6} + \frac{7 \times 2}{3 \times 2} \)

\( E(X) = \frac{2}{6} + \frac{3}{6} + \frac{5}{6} + \frac{14}{6} \)

\( E(X) = \frac{2 + 3 + 5 + 14}{6} \)

\( E(X) = \frac{24}{6} \)

\( E(X) = 4 \)

So, the expected value of \( X \) is 4.

Calculating the Expected Value of X Squared E(X²)

Next, we need to calculate \( E(X^2) \). This is the expected value of the square of the random variable. The formula is:

\( E(X^2) = \sum x_i^2 P(x_i) \)

First, square each value of \( X \):

• \( 1^2 = 1 \)
• \( 3^2 = 9 \)
• \( 5^2 = 25 \)
• \( 7^2 = 49 \)

Now, calculate \( E(X^2) \) using the squared values and their probabilities:

\( E(X^2) = (1^2 \times P(X=1)) + (3^2 \times P(X=3)) + (5^2 \times P(X=5)) + (7^2 \times P(X=7)) \)

\( E(X^2) = \left(1 \times \frac{1}{3}\right) + \left(9 \times \frac{1}{6}\right) + \left(25 \times \frac{1}{6}\right) + \left(49 \times \frac{1}{3}\right) \)

\( E(X^2) = \frac{1}{3} + \frac{9}{6} + \frac{25}{6} + \frac{49}{3} \)

Simplify the fraction \( \frac{9}{6} = \frac{3}{2} \). Find a common denominator for all terms, which is 6:

\( E(X^2) = \frac{1 \times 2}{3 \times 2} + \frac{9}{6} + \frac{25}{6} + \frac{49 \times 2}{3 \times 2} \)

\( E(X^2) = \frac{2}{6} + \frac{9}{6} + \frac{25}{6} + \frac{98}{6} \)

\( E(X^2) = \frac{2 + 9 + 25 + 98}{6} \)

\( E(X^2) = \frac{134}{6} \)

Simplify the fraction by dividing both numerator and denominator by 2:

\( E(X^2) = \frac{67}{3} \)

So, the expected value of \( X^2 \) is \( \frac{67}{3} \).

Applying the Variance Formula

Now that we have \( E(X) \) and \( E(X^2) \), we can calculate the variance \( Var(X) \) using the formula:

\( Var(X) = E(X^2) - [E(X)]^2 \)

Substitute the calculated values:

\( Var(X) = \frac{67}{3} - (4)^2 \)

\( Var(X) = \frac{67}{3} - 16 \)

To subtract the whole number 16 from the fraction \( \frac{67}{3} \), express 16 as a fraction with a denominator of 3:

\( 16 = \frac{16 \times 3}{3} = \frac{48}{3} \)

Now perform the subtraction:

\( Var(X) = \frac{67}{3} - \frac{48}{3} \)

\( Var(X) = \frac{67 - 48}{3} \)

\( Var(X) = \frac{19}{3} \)

Final Variance Result

The variance of the random variable \( X \) is \( \frac{19}{3} \).

Revision Table: Key Probability Concepts

Additional Information on Variance and Standard Deviation

The variance is a key measure of the dispersion or spread of a probability distribution. A higher variance indicates that the values of the random variable are more spread out from the mean, while a lower variance indicates they are clustered closer to the mean.

The standard deviation is the square root of the variance. It is often used because it has the same units as the random variable itself, making it easier to interpret the spread in the context of the problem.

\( \text{Standard Deviation} = \sigma = \sqrt{Var(X)} \)

In this specific problem, the standard deviation would be \( \sqrt{\frac{19}{3}} \).

Understanding how to calculate expected value and variance from a probability distribution is fundamental in probability and statistics and is often tested in exams.