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The given probability density function (pdf) of \( X \) is:

\[
f(x) = 
\begin{cases} 
\frac{1}{\lambda} e^{-\frac{x - 3}{\lambda}}, & x \geq 3 \\
0, & \text{otherwise}
\end{cases}
\]

This is an exponential distribution shifted by 3. For an exponential distribution of the form \( f(x) = \frac{1}{\lambda} e^{-\frac{x}{\lambda}} \), the **mode** is the lower bound (here 3), and the **median** is given by:

\[
\text{Median} = 3 + \lambda \ln(2)
\]

### Step-by-step:
1. **Mode of \( X \)**: Since this is an exponential distribution, the mode is simply the lower bound, which is 3.
   
2. **Median of \( X \)**: The median of an exponential distribution shifted by 3 is:
   \[
   \text{Median of } X = 3 + \lambda \ln(2)
   \]

3. **Expression in the problem**: The problem asks for the value of:

   \[
   \frac{1}{\lambda} \left( \text{Median of } X - \text{Mode of } X \right)
   \]
   Substituting the values:
   \[
   \frac{1}{\lambda} \left( (3 + \lambda \ln(2)) - 3 \right) = \frac{1}{\lambda} (\lambda \ln(2)) = \ln(2)
   \]

Thus, the value of the expression is \( \ln(2) \), and the correct answer is:

\[
\boxed{A. \ln2}
\]

---

Would you like more details on any part of the solution, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How do we derive the median of an exponential distribution?
2. What are the key properties of an exponential distribution?
3. How do we compute the mean and variance of an exponential distribution?
4. How do transformations of random variables affect their distributions?
5. Can you explain why the mode of the exponential distribution is the lower bound?

**Tip**: For exponential distributions, remember that the mode is always at the starting point of the distribution, and the median can be found using \( \lambda \ln(2) \).

Let X be a random variable with pdf: f(x) = (1/λ) * e^(-(x-3)/λ) for x ≥ 3, and 0 otherwise. Then value of (1/λ) * (median of X - mode of X) equals to ______. Options: A) ln2, B) ln4, C) ln3, D) None of these.

Solve the problem of finding the value of (1/λ) * (median of X - mode of X) for a shifted exponential distribution. This includes key exponential distribution properties and uses the median formula for shifted exponential distributions. Suitable for undergraduate-level statistics students.

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