1. Given distribution

A Poisson random variable with mean $2$ has parameter $\lambda = 2.$ So, $P(X=k)=\frac{e^{-2}2^k}{k!}, \qquad k=0,1,2,\dots$

2. Interpret the event $X>1.5$

Since $X$ is a discrete random variable taking only integer values, $X>1.5 \iff X\ge 2.$

Therefore, $P(X>1.5)=P(X\ge 2).$

3. Use complement

$P(X\ge 2)=1-P(X=0)-P(X=1).$

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

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

So, $P(X\ge 2)=1-e^{-2}-2e^{-2}=1-3e^{-2}=1-\frac{3}{e^2}.$

4. Match with the options

The value is $1-\frac{3}{e^2},$ which corresponds to Option C.

5. Comparison with stored answer

Stored correct answer: C

This matches our derived answer.