
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Find the point estimates of the unknown parameters \(\lambda_{1}\) and \(\lambda_{2}\) of the "double Poisson distribution" using the method of moments from the sample \(x_{1}, x_{2}, \ldots, x_{n}\).

$$
P\left(X=x_{i}\right)=\frac{1}{2} \cdot \frac{\lambda_{1}^{x_{i}} e^{-\lambda_{1}}}{x_{i}!}+\frac{1}{2} \cdot \frac{\lambda_{2}^{x_{i}} e^{-\lambda_{2}}}{x_{i}!}
$$

where \(x_{i}\) is the number of occurrences of the event in \(n_{i}\) trials, \(\lambda_{1}\) and \(\lambda_{2}\) are positive numbers, and \(\lambda_{2}>\lambda_{1}\).

Present the answer in LaTex format: \boxed{Your answer}
user