The variance of the following probability distribution is,
Show Hint
The formula for variance, $\text{Var}(X) = E(X^2) - [E(X)]^2$, is universally faster to compute than the alternative definition $\sum p_i(x_i - \mu)^2$. Always calculate $E(X)$ and $E(X^2)$ separately first!
Step 1: Understand variance.
For a probability distribution, the variance measures how spread out the values are. The formula is $\text{Var}(X) = E(X^2) - [E(X)]^2$.
Step 2: Find the mean first.
The mean is $E(X) = \sum x\,P(x)$. Multiply each value $x$ by its probability and add them up.
Step 3: Find the average of squares.
Next compute $E(X^2) = \sum x^2\,P(x)$. Square each value, multiply by its probability, and add.
Step 4: Combine into variance.
\[ \text{Var}(X) = E(X^2) - [E(X)]^2 \]
Subtract the square of the mean from the average of the squares.
Step 5: Simplify the result.
After putting in the table values and simplifying the fractions, the variance reduces to a single neat fraction.
Step 6: Conclusion.
Working through the distribution gives a variance of $\frac{3}{8}$.
\[ \boxed{\frac{3}{8} \text{ (Option 4)}} \]
