I have this statement:
Justify if it is correct or not, perform the following transformation:
Let $X$ a random var, thus:
$P = E(X) - X + 1$
$P = E(1 * X) - X + 1$
$P = X[E(1)] - X + 1$
With the property, $E(c) = c$, then:
$P = X*1 - X +1, => X - X + 1 = 1$
I think the trap step is to get X out of the expected value, but I do not know the explanation of this. Thanks in advance.
This is not correct since the linearity of the expected value (which is just an integral) is just the "constant rule": For any real random variable $X$ with finite expectancy, real constant $c$:
$$\Bbb E(c \cdot X) = \int_\Omega c \cdot X \,\mathrm d \Bbb P= c \int_\Omega X \,\mathrm d \Bbb P = c\cdot \Bbb E(X).$$
Can you see why the above steps fail (in general) if we replace $c$ by another random variable $Y$?
About your particular "riddle": Just consider $X\sim \mathcal N(0,1)$ - a standard normal distribution. Then $\Bbb E(X) - X + 1$ is simply $1-X\sim 1+X\sim \mathcal N(1,1)$, which doesn't equal $0$.