Problem
The American Association of Individual Investors publishes an annual guide to the opt mutual funds. The following table contains their ratings of the total risk and the probabilities. The value of $x=1$ is for low risk and $x=5$ for high risk. \begin{tabular}{|l|l|} \hline$X$ & $P(x)$ \\ \hline 1 & 0.05 \\ \hline 2 & 0.10 \\ \hline 3 & 0.15 \\ \hline 4 & 0.4 \\ \hline 5 & 0.3 \\ \hline \end{tabular} i. Calculate the probability that the risk level is more than 3. ii. What is the mean level of the risk? iii. What are the variance and the standard deviation of the level of the risk?
Solution
Step 1 ：Given the probabilities for each risk level are as follows: \(P(1) = 0.05\), \(P(2) = 0.10\), \(P(3) = 0.15\), \(P(4) = 0.4\), and \(P(5) = 0.3\).
Step 2 ：To calculate the probability that the risk level is more than 3, we need to add up the probabilities for the risk levels 4 and 5.
Step 3 ：So, the probability that the risk level is more than 3 is \(P(4) + P(5) = 0.4 + 0.3 = 0.7\).
Step 4 ：Final Answer: The probability that the risk level is more than 3 is \(\boxed{0.7}\).
