Two continuous random variables, X and Y, are related such that Y = aX + 5 where a is a positive integer and E(aX + 5) = Var(aX + 5). The mean of X is 9 and the variance of X is 2. Determine:
a. the value of the constant a b. E(Y) and Var(Y).

Given the problem, we have the following information:

- \( Y = aX + 5 \) where \( a \) is a positive integer.
- \( E(aX + 5) = \text{Var}(aX + 5) \).
- The mean of \( X \), \( E(X) = 9 \).
- The variance of \( X \), \( \text{Var}(X) = 2 \).

### Part (a): Determine the value of the constant \( a \)

Let's first calculate the expected value and variance of \( Y = aX + 5 \):

1. **Expected value of \( Y \):**
   \[
   E(Y) = E(aX + 5) = aE(X) + 5 = a \times 9 + 5 = 9a + 5
   \]

2. **Variance of \( Y \):**
   \[
   \text{Var}(Y) = \text{Var}(aX + 5)
   \]
   Since adding a constant does not affect the variance:
   \[
   \text{Var}(Y) = \text{Var}(aX) = a^2 \text{Var}(X) = a^2 \times 2 = 2a^2
   \]

Given that \( E(Y) = \text{Var}(Y) \), we set up the equation:

\[
9a + 5 = 2a^2
\]

Rearranging this into a standard quadratic equation form:

\[
2a^2 - 9a - 5 = 0
\]

We solve this quadratic equation using the quadratic formula:

\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 2 \), \( b = -9 \), and \( c = -5 \):

\[
a = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(2)(-5)}}{2(2)} = \frac{9 \pm \sqrt{81 + 40}}{4} = \frac{9 \pm \sqrt{121}}{4} = \frac{9 \pm 11}{4}
\]

This gives us two solutions:

\[
a = \frac{20}{4} = 5 \quad \text{or} \quad a = \frac{-2}{4} = -0.5
\]

Since \( a \) is a positive integer, \( a = 5 \) is the valid solution.

### Part (b): Calculate \( E(Y) \) and \( \text{Var}(Y) \)

Now that we know \( a = 5 \), we can find \( E(Y) \) and \( \text{Var}(Y) \).

1. **Expected value of \( Y \):**
   \[
   E(Y) = 9a + 5 = 9(5) + 5 = 45 + 5 = 50
   \]

2. **Variance of \( Y \):**
   \[
   \text{Var}(Y) = 2a^2 = 2(5^2) = 2 \times 25 = 50
   \]

Thus, the answers are:
- **(a)** \( a = 5 \)
- **(b)** \( E(Y) = 50 \) and \( \text{Var}(Y) = 50 \)

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How do you find the expected value and variance of a linear transformation of a random variable?
2. What are some other types of problems involving continuous random variables?
3. How does adding or multiplying by constants affect the mean and variance of a random variable?
4. Can this method be applied to other forms of linear relationships between random variables?
5. What are the implications if \( a \) were not a positive integer?

**Tip:** Always double-check the assumptions and given conditions in probability problems, as they can significantly affect the solution approach.

Explore how to determine the expected value and variance of a continuous random variable transformed by a linear function. This problem involves calculating these statistics for Y = aX + 5, where X is a random variable with a given mean and variance. Suitable for advanced high school students and beyond.

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