A random variable $$X$$ has Poisson distribution with mean $$2$$. Then $$P(X > 1.5)$$ equals

The probability mass function of a Poisson random variable with mean (or parameter) $$\lambda$$ is

$$P(X = k) = \frac{\lambda^{\,k}}{k!}\,e^{-\lambda}, \quad k = 0,1,2,\dots$$

Here $$\lambda = 2$$. The event $$X \gt 1.5$$ is the same as $$X \ge 2$$ because $$X$$ takes only non-negative integer values.

Hence

$$P(X \gt 1.5) = P(X \ge 2) = 1 - P(X \le 1) = 1 - \bigl[P(X = 0) + P(X = 1)\bigr].$$

Compute each term using the Poisson formula:

$$P(X = 0) = \frac{2^{\,0}}{0!}\,e^{-2} = e^{-2},$$
$$P(X = 1) = \frac{2^{\,1}}{1!}\,e^{-2} = 2e^{-2}.$$

Therefore

$$P(X \le 1) = e^{-2} + 2e^{-2} = 3e^{-2}.$$

Finally,

$$P(X \gt 1.5) = 1 - 3e^{-2} = 1 - \frac{3}{e^{2}}.$$

Option C which is: $$1 - \frac{3}{e^2}$$