[convolution] convolution with probit and Gaussian functions

https://stats.stackexchange.com/questions/61080/how-can-i-calculate-int-infty-infty-phi-left-fracw-ab-right-phiw

Let x [from normal dist. N(a, b2)] and y [from normal dist. N(1, 0)] be independent random variables

Then,

𝑃{𝑋≤𝑌∣𝑌=𝑤}=𝑃{𝑋≤𝑤}=Φ(𝑤−𝑎𝑏).

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$$$$So, we obtain

𝑃{𝑋≤𝑌}=∫∞−∞𝑃{𝑋≤𝑌∣𝑌=𝑤}𝜙(𝑤)d𝑤=∫∞−∞Φ(𝑤−𝑎𝑏)𝜙(𝑤)d𝑤.$$$$
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By  2 Gaussian random variables 𝑋$X$ and 𝑌$Y$ and consider 𝑋<𝑌$X<Y$ is equivalent to 𝑋−𝑌<0

By considering the condition x<=y as x-y<=0, 

𝑃{𝑋≤𝑌}=𝑃{𝑋−𝑌≤0} can be expressed in terms of Φ(⋅)$\mathrm{\Phi }(\cdot )$ 

∫∞−∞Φ(𝑤−𝑎𝑏)𝜙(𝑤)d𝑤=Φ(−𝑎𝑏2+1‾‾‾‾‾‾√)