To find the value of $k$ for the given probability density function (PDF) $f(x)$, we need to use the property that the total area under the PDF curve must equal 1. This is a fundamental requirement for all probability density functions.

Given:

$f(x) = k \text{ for } 0 \leq x \leq 4$ $f(x) = 0 \text{ otherwise}$

Step 1: Set up the integral of the PDF over its entire range.

The total probability is given by the integral of $f(x)$ over the interval from 0 to 4: $\int_{0}^{4} f(x) \, dx = 1$

Since $f(x) = k$ over this interval, the equation becomes: $\int_{0}^{4} k \, dx = 1$

Step 2: Solve the integral.

The integral of a constant $k$ over the interval from 0 to 4 is: $k \int_{0}^{4} 1 \, dx = k \times [x]_{0}^{4} = k \times (4 - 0) = 4k$

So we have: $4k = 1$

Step 3: Solve for $k$.

$k = \frac{1}{4}$

Conclusion:

The value of $k$ for the given probability density function is $\frac{1}{4}$.

---

Would you like more details or have any questions? Here are some related questions:

1. What is a probability density function (PDF)?
2. How do you verify if a function is a valid PDF?
3. What is the difference between discrete and continuous random variables?
4. Can you explain the concept of cumulative distribution function (CDF)?
5. How is the expected value of a continuous random variable calculated?
6. What is the role of normalization in probability density functions?
7. How do you find the median of a continuous random variable?
8. What is the significance of the area under the curve in a PDF?

Tip: Always remember that the integral of a PDF over its entire range must equal 1, ensuring that the total probability is properly normalized.