Showing $\Phi(a+\text{weight}\cdot b) \neq (1-\text{weight})\cdot\Phi(a)+\text{weight}\cdot \Phi(a+b)$ where $\Phi(\cdot)$ is the CDF of $N(0,1)$

Question

I came into a problem to prove the following inequality, but I couldn't figure it out. Can some one help?

$$\Phi(a+\text{weight}\cdot b) \neq (1-\text{weight})\cdot\Phi(a)+\text{weight}\cdot \Phi(a+b)$$, where $\Phi(\cdot)$ is the CDF function of a standard normal distribution.

What would be appropriate way to prove it? Would take derivative work?

Thanks!

Answer 1

You can think about it this way:

Fix $a,b>0$ and consider the interval $[a, a+b]$. Suppose your statement holds.

If you look at the RHS, you can see that it is a linear function of the weight and in particular, is a line between $\Phi(a)$ and $\Phi(a+b)$. The LHS, however, is not a linear function as this CDF of a normal distribution is nowhere linear.

You can use similar arguments when some of $a,b<0$ by just changing the interval.