A random variable X is defined by the transformation Z = log(X), where the mean of the random variable Z is E(Z) = 0. Is E(X) greater than, less than, or equal to 1?
I know we need to apply Jensens Inequality, And I know that $X= e^Z$. But am getting really confused on what is E(X).
Jensen's inequality should be thought of independently of expectations and probabilities.
It's just a statement that the weighted average of a convex function evaluated at a number of points is at least as great as the function evaluated at the weighted average of the points. So for a convex function $f$, this means that$$ w_1 \, f(x_1) + w_2 \, f(x_2) + \ldots + w_n \, f(x_n) \ge f(w_1 x_1 + w_2 x_2 + \ldots + w_n x_n) $$ where $$ w_i \ge 0 \text{ and } \sum_{i=1}^n w_i = 1. $$
If you pick $n=2$, you can see this immediately by noticing that a convex function always lies below the line connecting any two of its points
Tying back to probabilities and expectations, notice that expectations are weighted averages, and that the function $f(x) = \exp(x)$ is convex. So the expectation of $f(x)$ will be greater than $f( \text{expectation}(x) )$.