by RAHUl JHa
Last Updated January 16, 2018 18:19 PM

Consider a random variable with the probability density function as $f(x)=2\varphi (x)\phi(x)$, $x$ is a real number, and $\varphi(x)$ is the density function of Standard Normal distribution, $N(0,1)$ while $\phi(x)$ is the cumulative distribution function of Standard Normal distribution, then which of the following will be true:

- $E(x)>0$
- $E(x)<0$
- $P(X\leq0)>0.5$
- $P(X\geq0)>0.25$

I have no idea how to proceed with the problem. Anything which can make me begin this problem will be helpful. I am totally stuck at this. Thanks in advance.

Based on the relationship $\varphi(x) = \frac{d}{dx} \, \phi(x)$, we can obtain the CDF of this density through integration (chain rule). The CDF will be able to tell us how distorted this new distribution is relative to the normal CDF.

If you get stuck on the integration or anything, just let me know.

Hint for the last two: use integration by parts to show that $P(X \le 0) = .5 - P(X \le 0)$.

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