# What' s $E[Y] = E[f(X)]$? $X\sim N(\mu,\sigma^2)$ and $f()$ is cdf of std normal rv? [duplicate]

What's the expected value of $$Y= \Phi(X)$$ where $$X$$ is a normal random variable with mean $$\mu$$ and variance $$\sigma^2$$ and $$\Phi$$ being the cdf of a standard normal distribution?

• Hi: What is really being calculated is the average of the value under the standardized normal density ( the bell curve ). So, it's not as difficult as it might first look when viewed from that perspective. Does that help you ? Nov 9, 2020 at 1:18
• Sorry, I still find it a bit hard to understand. The X is a normal random variable with mean 𝜇 and variance 𝜎2, not a standardized one. Although, intuitively if X were standardized rv, the expected value of Y looks like 1/2, right? Nov 10, 2020 at 5:25
• I got stuff going on at the moment but I just quickly glanced at the answers ( assuming the questions are the same. I'm not totally sure about that ) and sounds like it might be as difficult as it first looks :). Maybe someone else in can chime in and comment on whether they are the same ? I think what you said is on the right track but maybe not correct given my glance at the answers. I'm hoping whuber can comment or answer since he answered the one that is supposedly the same question. whuber: thanks in advance whether you can say something or not. Nov 10, 2020 at 14:12