Consider a random variable to which a Poisson distribution is best fitted. It happens that P(x = 1) = $$\frac{2}{3}$$P(x = 2) on this distribution plot. The variance of this distribution will be A. 3 B. 2 C. 1 D. $$\frac{2}{3}$$

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 $\frac{2}{3}$.