Question 1207673
To analyze the total risk ratings for mutual funds, we will follow the steps outlined in the question.

### Step 1: Develop a Probability Distribution for Level of Risk

#### a) Assign Values and Calculate Probabilities

We will assign values to the risk categories as follows:
- Low risk: $ x_1 = 1 $
- Below Average risk: $ x_2 = 2 $
- Average risk: $ x_3 = 3 $
- Above Average risk: $ x_4 = 4 $
- High risk: $ x_5 = 5 $

Next, we will calculate the total number of fund categories and the probability for each risk level.

**Total Number of Fund Categories**:
$$
\text{Total} = 7 + 6 + 3 + 6 + 7 = 29
$$

**Probability Distribution**:
$$
P(x_1) = \frac{7}{29}, \quad P(x_2) = \frac{6}{29}, \quad P(x_3) = \frac{3}{29}, \quad P(x_4) = \frac{6}{29}, \quad P(x_5) = \frac{7}{29}
$$

The probability distribution can be summarized as follows:

$$
\begin{array}{|c|c|c|}
\hline
\text{Risk Level} & \text{Value (x)} & \text{Probability (P)} \\
\hline
\text{Low} & 1 & \frac{7}{29} \\
\text{Below Average} & 2 & \frac{6}{29} \\
\text{Average} & 3 & \frac{3}{29} \\
\text{Above Average} & 4 & \frac{6}{29} \\
\text{High} & 5 & \frac{7}{29} \\
\hline
\end{array}
$$

### Step 2: Calculate the Expected Value and Variance

#### b) Expected Value $ E(X) $

The expected value $ E(X) $ is calculated as follows:

$$
E(X) = \sum (x_i \cdot P(x_i))
$$

Calculating each term:

$$
E(X) = 1 \cdot \frac{7}{29} + 2 \cdot \frac{6}{29} + 3 \cdot \frac{3}{29} + 4 \cdot \frac{6}{29} + 5 \cdot \frac{7}{29}
$$

Calculating each term:

$$
E(X) = \frac{7}{29} + \frac{12}{29} + \frac{9}{29} + \frac{24}{29} + \frac{35}{29}
$$

Summing these values:

$$
E(X) = \frac{7 + 12 + 9 + 24 + 35}{29} = \frac{87}{29} \approx 3
$$

#### c) Variance $ Var(X) $

The variance $ Var(X) $ is calculated using the formula:

$$
Var(X) = E(X^2) - (E(X))^2
$$

First, we need to calculate $ E(X^2) $:

$$
E(X^2) = \sum (x_i^2 \cdot P(x_i))
$$

Calculating each term:

$$
E(X^2) = 1^2 \cdot \frac{7}{29} + 2^2 \cdot \frac{6}{29} + 3^2 \cdot \frac{3}{29} + 4^2 \cdot \frac{6}{29} + 5^2 \cdot \frac{7}{29}
$$

Calculating each term:

$$
E(X^2) = \frac{7}{29} + \frac{24}{29} + \frac{27}{29} + \frac{96}{29} + \frac{175}{29}
$$

Summing these values:

$$
E(X^2) = \frac{7 + 24 + 27 + 96 + 175}{29} = \frac{329}{29} \approx 11.34
$$

Now, we can calculate the variance:

$$
Var(X) = E(X^2) - (E(X))^2 = \frac{329}{29} - \left(\frac{87}{29}\right)^2
$$

Calculating $ (E(X))^2 $:

$$
(E(X))^2 = \left(\frac{87}{29}\right)^2 = \frac{7569}{841}
$$

Now, we need to convert $ E(X^2) $ to have a common denominator:

$$
E(X^2) = \frac{329}{29} = \frac{329 \times 29}{29 \times 29} = \frac{9541}{841}
$$

Now we can calculate the variance:

$$
Var(X) = \frac{9541}{841} - \frac{7569}{841} = \frac{1972}{841} \approx 2.34
$$

### Step 3: Compare Total Risk of Bond Funds with Stock Funds

#### c) Bond Funds Analysis

From the information provided:
- Total bond fund categories: 11
- Low risk bond funds: 7
- Below average bond funds: 4

**Risk Distribution for Bond Funds**:
- Low: 7
- Below Average: 4
- Average: 0
- Above Average: 0
- High: 0

**Probability Distribution for Bond Funds**:
$$
\begin{array}{|c|c|c|}
\hline
\text{Risk Level} & \text{Value (x)} & \text{Probability (P)} \\
\hline
\text{Low} & 1 & \frac{7}{11} \\
\text{Below Average} & 2 & \frac{4}{11} \\
\text{Average} & 3 & 0 \\
\text{Above Average} & 4 & 0 \\
\text{High} & 5 & 0 \\
\hline
\end{array}
$$

**Risk Distribution for Stock Funds**:
- Total stock fund categories: $ 29 - 11 = 18 $
- Low: $ 7 $
- Below Average: $ 6 $
- Average: $ 3 $
- Above Average: $ 6 $
- High: $ 7 $

**Probability Distribution for Stock Funds**:
$$
\begin{array}{|c|c|c|}
\hline
\text{Risk Level} & \text{Value (x)} & \text{Probability (P)} \\
\hline
\text{Low} & 1 & \frac{7}{18} \\
\text{Below Average} & 2 & \frac{6}{18} \\
\text{Average} & 3 & \frac{3}{18} \\
\text{Above Average} & 4 & \frac{6}{18} \\
\text{High} & 5 & \frac{7}{18} \\
\hline
\end{array}
$$

### Summary of Findings

- **Expected Value for Total Risk**: $ E(X) \approx 3 $
- **Variance for Total Risk**: $ Var(X) \approx 2.34 $
- **Bond Funds**: Higher concentration in low risk (7 out of 11) compared to stock funds.
- **Stock Funds**: More diverse risk distribution across categories.

This analysis shows that bond funds tend to be rated lower in risk compared to stock funds, which have a wider range of risk ratings.