Let's address the problem systematically:

---

(a) Find the MLE for $\mu$

The given pmf is: $f_X(x; \mu) = e^{-\mu x} \frac{(\mu x)^{x-1}}{x!}, \quad x \in \{1, 2, \ldots\}, \, \mu \in (0, 1).$

For a set of iid observations $X_1, X_2, \ldots, X_n$, the likelihood function is: $L(\mu; X_1, \ldots, X_n) = \prod_{i=1}^n f_X(X_i; \mu).$

Substituting $f_X(x; \mu)$: $L(\mu; X_1, \ldots, X_n) = \prod_{i=1}^n e^{-\mu X_i} \frac{(\mu X_i)^{X_i - 1}}{X_i!}.$

Simplifying: $L(\mu; X_1, \ldots, X_n) = e^{-\mu \sum_{i=1}^n X_i} \prod_{i=1}^n \frac{(\mu X_i)^{X_i - 1}}{X_i!}.$

Taking the log-likelihood: $\ell(\mu) = \log L(\mu; X_1, \ldots, X_n) = -\mu \sum_{i=1}^n X_i + \sum_{i=1}^n (X_i - 1) \log (\mu X_i) - \sum_{i=1}^n \log (X_i!).$

Focus on terms involving $\mu$: \ell(\mu) = -\mu \sum_{i=1}^n X_i + \sum_{i=1}^n (X_i - 1) \log \mu + \text{(terms independent of \mu)}.

Differentiating with respect to $\mu$ and setting the derivative to zero: $\frac{\partial \ell}{\partial \mu} = -\sum_{i=1}^n X_i + \sum_{i=1}^n \frac{X_i - 1}{\mu} = 0.$

Simplify: $\mu \sum_{i=1}^n X_i = \sum_{i=1}^n (X_i - 1).$

Rearrange: $\mu = \frac{\sum_{i=1}^n (X_i - 1)}{\sum_{i=1}^n X_i}.$

---

For $x_1 = 2, x_2 = 1, x_3 = 6$:

$\sum_{i=1}^3 (X_i - 1) = (2 - 1) + (1 - 1) + (6 - 1) = 1 + 0 + 5 = 6,$ $\sum_{i=1}^3 X_i = 2 + 1 + 6 = 9.$

Thus: $\mu = \frac{6}{9} = \frac{2}{3}.$

---

(b) Graph the Likelihood Function

The likelihood function is: $L(\mu) = e^{-\mu \cdot 9} \cdot \frac{(\mu \cdot 2)^{1} \cdot (\mu \cdot 1)^{0} \cdot (\mu \cdot 6)^{5}}{2! \cdot 1! \cdot 6!}.$

Simplify constants: $L(\mu) = C \cdot e^{-9\mu} \cdot (\mu)^{1 + 0 + 5},$ where $C$ is independent of $\mu$.

Explicitly: $L(\mu) \propto \mu^6 e^{-9\mu}.$

---

Plot on Desmos

1. Define $L(\mu) = \mu^6 e^{-9\mu}$.
2. Choose a range for $\mu$ (e.g., $\mu \in (0, 1)$).
3. Observe the maximum near $\mu = \frac{2}{3}$.

Would you like help generating or uploading the graph? I can also generate the graph for you if needed.

---

Follow-Up Questions:

1. Why is MLE an important method for parameter estimation in statistics?
2. What are the practical interpretations of $\mu$ in this context?
3. How does the likelihood function differ from the log-likelihood function?
4. What are the benefits of using Desmos for visualization in statistical analysis?
5. What conditions ensure the uniqueness of the MLE in this type of problem?

Tip: Always validate the MLE by checking the second derivative to confirm it is a maximum, not a minimum or inflection point.