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: Understanding the Question:
We are asked to find the variance of a discrete probability distribution. An image of a table containing random variable values ($x_i$) and their probabilities ($p_i$) is referenced.
Step 2: Key Formula or Approach:
The variance of a discrete random variable $X$ is calculated using the formula:
$$\text{Variance}(X) = \sum p_i x_i^2 - \left(\sum p_i x_i\right)^2$$
where $\sum p_i x_i^2$ is the expected value of $X^2$, and $\sum p_i x_i$ is the expected value of $X$ (the mean).
Step 3: Detailed Explanation:
From the context of the solution, the sums have been evaluated as follows:
$$\sum p_i x_i^2 = \frac{5}{8}$$
$$\sum p_i x_i = \frac{1}{2}$$
Substitute these values directly into the variance formula:
$$\text{Variance} = \frac{5}{8} - \left(\frac{1}{2}\right)^2$$
Square the mean term:
$$\text{Variance} = \frac{5}{8} - \frac{1}{4}$$
To subtract, find a common denominator (which is 8):
$$\text{Variance} = \frac{5}{8} - \frac{2}{8} = \frac{3}{8}$$
Step 4: Final Answer:
The variance of the distribution is $\frac{3}{8}$, matching option (D).
