Find k so that the function f(x) defined by
`f(x) = ke^(-3x), x gt 0`
= 0 , otherwise.
is a probability density function.

Answer

To find the value of \( k \) such that the function

\[ f(x) = \begin{cases} ke^{-3x} & \text{if } x > 0 \\ 0 & \text{otherwise} \end{cases} \]

is a probability density function (pdf), we need to ensure that the total area under the curve of \( f(x) \) over its entire range is equal to 1. This means we need to solve the following integral:

\[ \int_{0}^{\infty} f(x) \, dx = 1 \]

### Step 1: Set up the integral

We substitute \( f(x) \) into the integral:

\[ \int_{0}^{\infty} ke^{-3x} \, dx = 1 \]

### Step 2: Factor out \( k \)

Since \( k \) is a constant, we can factor it out of the integral:

\[ k \int_{0}^{\infty} e^{-3x} \, dx = 1 \]

### Step 3: Evaluate the integral

Now we need to evaluate the integral \( \int_{0}^{\infty} e^{-3x} \, dx \). This is a standard integral and can be computed as follows:

\[ \int e^{-3x} \, dx = -\frac{1}{3} e^{-3x} \]

Now we evaluate it from 0 to \( \infty \):

\[ \left[-\frac{1}{3} e^{-3x}\right]_{0}^{\infty} = \left(0 - \left(-\frac{1}{3}\right)\right) = \frac{1}{3} \]

### Step 4: Substitute back into the equation

Now we substitute this result back into our equation:

\[ k \cdot \frac{1}{3} = 1 \]

### Step 5: Solve for \( k \)

To find \( k \), we multiply both sides by 3:

\[ k = 3 \]

### Final Answer

Thus, the value of \( k \) is

\[ \boxed{3} \]