3
2
1
$$ rac{2}{3}$$
Answer is Right!
Answer is Wrong!
The correct answer is $\boxed{\text{B}}$.
A Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event.
The variance of a Poisson distribution is equal to its mean. In this case, we are given that $P(x = 1) = \frac{2}{3}P(x = 2)$. This means that the mean of the distribution is $\mu = 2$. The variance of the distribution is therefore $\sigma^2 = \mu = 2$.
Here is a brief explanation of each option:
- Option A: $3$. This is not the correct answer because the variance of a Poisson distribution is always less than or equal to its mean.
- Option B: $2$. This is the correct answer because the variance of a Poisson distribution is equal to its mean.
- Option C: $1$. This is not the correct answer because the variance of a Poisson distribution is always greater than or equal to $1$.
- Option D: $\frac{2}{3}$. This is not the correct answer because the variance of a Poisson distribution is always greater than or equal to $\frac{2}{3}$.