8. Solve for the variance \begin{tabular}{|c|c|} \hline $\boldsymbol{x}$ & $\boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x})$ \\ \hline-3 & 0.17 \\ \hline-1 & 0.28 \\ \hline 1 & 0.15 \\ \hline 3 & 0.27 \\ \hline 5 & 0.13 \\ \hline \end{tabular}
Solution
Step 1 ：Given the random variable X with the following values and their corresponding probabilities: \(-3, -1, 1, 3, 5\) with probabilities \(0.17, 0.28, 0.15, 0.27, 0.13\) respectively.
Step 2 ：The variance of a random variable X is given by the formula Var(X) = E(X^2) - [E(X)]^2. Here, E(X) is the expected value of X, and E(X^2) is the expected value of X^2.
Step 3 ：The expected value of a random variable is calculated by summing the product of each outcome and its probability. So, first, we need to calculate E(X) and E(X^2).
Step 4 ：Calculating E(X), we get \(E(X) = 0.8200000000000001\).
Step 5 ：Calculating E(X^2), we get \(E(X^2) = 7.640000000000001\).
Step 6 ：Substituting these values into the formula for variance, we get \(Var(X) = E(X^2) - [E(X)]^2 = 6.967600000000001\).
Step 7 ：Rounding to two decimal places, the variance of the random variable X is \(\boxed{6.97}\).
