Question

Let $X_{1}, X_{2}, \dots, X_{n}$ be random sample from a Poisson family with parameter $\lambda$. Then the maximum likelihood estimate of $P(X \ge 2)$ is

• A. $1 - \overline{X} e^{-\overline{X}}$
• B. $(1 + \overline{X}) e^{-\overline{X}}$
• C. $\overline{X} e^{-\overline{X}}$
• D. $1 - (1 + \overline{X}) e^{-\overline{X}}$

Hint:

The MLE invariance property is extremely powerful: simply find the MLE of the basic parameter and plug it into whatever formula or probability you are asked to estimate.

Answer 1

The Correct Option is D

To find the Maximum Likelihood Estimate (MLE) of a function of a parameter, we use the invariance property of MLEs.

Step 1: \color{redFind the MLE of the Parameter $\lambda$
For a Poisson distribution with parameter $\lambda$, the likelihood function is:
$L(\lambda) = \prod_{i=1}^n \frac{e^{-\lambda} \lambda^{x_i}}{x_i!} = \frac{e^{-n\lambda} \lambda^{\sum x_i}}{\prod x_i!}$.
Taking the log-likelihood and differentiating with respect to $\lambda$:
$\frac{d}{d\lambda} \ln L(\lambda) = -n + \frac{\sum x_i}{\lambda} = 0 \implies \hat{\lambda} = \frac{\sum x_i}{n} = \overline{X}$.

Step 2: \color{redDefine the Function to Estimate
We need the MLE of $g(\lambda) = P(X \ge 2)$.
Using the complement rule:
$P(X \ge 2) = 1 - P(X = 0) - P(X = 1)$.

Step 3: \color{redCalculate the Probability Function
For $X \sim Poisson(\lambda)$:
$P(X=0) = e^{-\lambda}$
$P(X=1) = \lambda e^{-\lambda}$
So, $g(\lambda) = 1 - e^{-\lambda} - \lambda e^{-\lambda} = 1 - (1 + \lambda) e^{-\lambda}$.

Step 4: \color{redApply the Invariance Property
The invariance property states that if $\hat{\lambda}$ is the MLE of $\lambda$, then $g(\hat{\lambda})$ is the MLE of $g(\lambda)$.
Substituting $\hat{\lambda} = \overline{X}$:
$\widehat{P(X \ge 2)} = 1 - (1 + \overline{X}) e^{-\overline{X}}$.
This matches Option (4).