If $N$ denotes the cumulative function for the standard normal distribution, i.e. $N(0)=0.5$, $N(0.5)=0.6915$ etc. are you able to say anything more generally about $N(a) + N(b)$ for example and also what about $\alpha N(a) + \beta N(b)$ where $a$, $b$, $\alpha$ and $\beta$ are any real numbers.
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$\begingroup$ what kind of generality are you thinking of? $\endgroup$– Christoph HanckMay 7, 2015 at 11:21
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5$\begingroup$ That's a distribution function that falls within [0,1]. I don't see that adding distribution functions has any statistical meaning. The sum of two distribution functions will vary between 0 and 2, but why is this of interest or use? $\endgroup$– Nick CoxMay 7, 2015 at 11:21
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I don't know what exactly is wanted here but with $\Phi(\cdot)$ denoting the cumulative probability distribution function of the standard normal random variable, we can say the following.
If $a, b > 0$, then $\Phi(a) + \Phi(b) \leq 2\Phi\left(\frac{a+b}{2}\right)$ with equality holding only when $a=b$.
If $a, b < 0$, then $\Phi(a) + \Phi(b) \geq 2\Phi\left(\frac{a+b}{2}\right)$ with equality holding only when $a=b$.
If $a < 0, b > 0$, then $\Phi(a) + \Phi(b)$ is greater than $1$ or less than $1$ according as $|a| < b$ or $|a| > b$ with equality only when $a = -b$.
answered May 8, 2015 at 16:57