For the probability distribution given by following
the value of $\text{Var}(X) =$
Show Hint
Always double check your value of $E(X)$. Since the distribution is heavily weighted around the values 7 and 8 (each having a probability of $0.3$), the mean must fall very close to $7.5$. Finding $E(X) = 7.3$ tells you that your preliminary arithmetic is well on track!
Step 1: Recall the probability rule. All probabilities in the table must add to 1, which lets us find the unknown $k$. Step 2: Solve for k. The known probabilities sum to $0.07+0.2+0.3+0.07+0.04+0.02 = 0.7$, so $0.7 + k = 1$ gives $k = 0.3$ (this is $P(X=8)$). Step 3: Write the variance formula. $\text{Var}(X) = E(X^2) - [E(X)]^2$, where $E(X) = \sum x_i p_i$ and $E(X^2) = \sum x_i^2 p_i$. Step 4: Compute the mean. $E(X) = 5(0.07)+6(0.2)+7(0.3)+8(0.3)+9(0.07)+10(0.04)+11(0.02) = 0.35+1.2+2.1+2.4+0.63+0.4+0.22 = 7.3$. Step 5: Compute the mean of squares. $E(X^2) = 25(0.07)+36(0.2)+49(0.3)+64(0.3)+81(0.07)+100(0.04)+121(0.02) = 1.75+7.2+14.7+19.2+5.67+4.0+2.42 = 54.94$. Step 6: Subtract to get the variance. $\text{Var}(X) = 54.94 - (7.3)^2 = 54.94 - 53.29 = 1.65$, which is option 3 and matches the key. \[ \boxed{\text{Var}(X) = 1.65} \]
