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The problem provides a probability density function (PDF) \( f(x) \) for a random variable \( X \). The function \( f(x) \) is piecewise:

\[
f(x) = 
\begin{cases} 
Kx & \text{for } 0 \leq x \leq 5 \\
K(x-3) & \text{for } 5 < x \leq 10 \\
0 & \text{otherwise}
\end{cases}
\]

The questions involve:

1. **Finding the value of \( K \)** using the property that the total probability over the entire range must equal 1.
2. **Expected value \( E(X) \)**.
3. **Variance \( \text{Var}(X) \)**.
4. **Probability \( P(4 \leq X \leq 6) \)**.
5. **Sketching the graph of \( f(x) \)**.

Let's solve each part:

### (a) Finding the value of \( K \)
To find \( K \), we use the property that the total area under the PDF must equal 1:

\[
\int_{-\infty}^{\infty} f(x) dx = 1
\]

This simplifies to calculating the area for the two intervals \( [0, 5] \) and \( [5, 10] \).

\[
\int_0^5 Kx \, dx + \int_5^{10} K(x-3) \, dx = 1
\]

Let me calculate \( K \) first.The value of \( K \) is \( \frac{1}{35} \).

### (b) Finding the expected value \( E(X) \)

The expected value \( E(X) \) is given by:

\[
E(X) = \int_{-\infty}^{\infty} x f(x) dx
\]

This will involve two integrals over the intervals \( [0, 5] \) and \( [5, 10] \):

\[
E(X) = \int_0^5 x \cdot \frac{1}{35} x \, dx + \int_5^{10} x \cdot \frac{1}{35} (x-3) \, dx
\]

Let me compute this next.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, here’s the approach to compute the expected value:

1. Set up the integral for \( E(X) \):
   \[
   E(X) = \int_0^5 x \cdot \frac{1}{35} x \, dx + \int_5^{10} x \cdot \frac{1}{35} (x-3) \, dx
   \]
2. Calculate each integral for the two regions and sum them to get \( E(X) \).

Would you like help with the next part, or have any questions about these steps? Here are some related questions:

1. How do we interpret the expected value in probability theory?
2. How is the variance \( \text{Var}(X) \) calculated after finding \( E(X) \)?
3. How do we compute probabilities over specific intervals like \( P(4 \leq X \leq 6) \)?
4. What is the importance of normalizing the probability density function (finding \( K \))?
5. How would the graph of \( f(x) \) look like based on the function definition?

**Tip**: When solving integrals of piecewise functions, remember to treat each piece independently over its specific interval.

Given a random variable X with the following PDF: f(X) = Kx for 0 <= x <= 5, K(x - 3) for 5 <= x <= 10, and 0 otherwise. Solve for the value of K, E(X), Var(X), P(4 <= X <= 6), and sketch the graph.

Solve for the normalization constant K, expected value, variance, and probability for the random variable X with a piecewise probability density function (PDF). Includes detailed steps for calculating E(X), Var(X), and the probability between 4 and 6.

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