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We are given a probability distribution table for a random variable \( X \), where \( a \) and \( b \) are real numbers, and the expected value \( E(X) = \frac{109}{30} \).

The table looks like this:

\[
\begin{array}{c|c|c|c|c}
x & -5 & 1 & 6 & 10 \\
\hline
P(X = x) & 2a & b - a & \frac{1}{6} & \frac{1}{3} \\
\end{array}
\]

We need to find \( a \) and \( b \) (part a), and then calculate the standard deviation \( \sigma(X) \) of the random variable \( X \) rounded to two decimal places (part b).

### Part (a): Determining \( a \) and \( b \)
The two conditions are:
1. The probabilities must sum to 1:  
   \[
   2a + (b - a) + \frac{1}{6} + \frac{1}{3} = 1
   \]
2. The expected value is given as \( E(X) = \frac{109}{30} \). The formula for the expected value is:  
   \[
   E(X) = (-5 \cdot 2a) + (1 \cdot (b - a)) + (6 \cdot \frac{1}{6}) + (10 \cdot \frac{1}{3}) = \frac{109}{30}
   \]
Using the first equation, we can solve for \( a \) and \( b \).

Let me calculate these values.The solution to the system of equations is:

- \( a = \frac{1}{10} \)
- \( b = \frac{4}{10} = 0.4 \)

This matches the given partial result for \( a \).

### Part (b): Calculating the Standard Deviation \( \sigma(X) \)

The standard deviation is calculated using the formula:

\[
\sigma(X) = \sqrt{E(X^2) - (E(X))^2}
\]

Where:
- \( E(X^2) \) is the expected value of \( X^2 \), which can be computed as:
\[
E(X^2) = (-5)^2 \cdot P(X = -5) + 1^2 \cdot P(X = 1) + 6^2 \cdot P(X = 6) + 10^2 \cdot P(X = 10)
\]

Let me now calculate the standard deviation.The standard deviation \( \sigma(X) \) of the random variable \( X \) is approximately \( 5.61 \), rounded to two decimal places.

Do you have any further questions or would you like more details on the process?

Here are 5 related questions to expand on this topic:
1. How is the expected value \( E(X) \) calculated for a discrete random variable?
2. What does the variance represent in a probability distribution?
3. Can you derive the formula for standard deviation from the variance?
4. How do different probability distributions affect the expected value and variance?
5. What are the key differences between standard deviation and mean absolute deviation?

**Tip:** When calculating the variance and standard deviation, ensure that you calculate \( E(X^2) \) before subtracting \( (E(X))^2 \).

Given is a probability distribution of the random variable X, where a and b are real numbers. The expected value E(X) is 109/30. The table contains the following values for X: -5, 1, 6, and 10 with corresponding probabilities: 2a, b - a, 1/6, and 1/3 respectively. Determine a and b. [Hint: a = 1/10] (Part a). Calculate the standard deviation of X rounded to two decimal places (Part b).

This problem involves calculating the unknown parameters 'a' and 'b' in a given probability distribution with an expected value of 109/30. Additionally, the task includes computing the standard deviation of the random variable X, which takes values -5, 1, 6, and 10. A step-by-step solution is provided, ideal for college-level students.

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