Question: The probability distributions for 2 variables are defined as follows$ X$ ~ $N$$(120, σ^2)$ and $Y$ ~ $N$$(μ, 2σ^2)$ and $P(X < 124)$ = $P(Y > 124$). Calculate $μ$.
I tried this for hours and no result. Since there are 2 unknowns, I tried forming a simultaneous equation. Z-scores?
Let $W:=(X-120)/\sigma,\,Z:=(Y-\mu)/(\sigma\sqrt{2})$ so $Z,\,W\sim N(0,\,1)$ and$$P(W<4/\sigma)=P(Z>(124-\mu)/(\sigma\sqrt{2}))=P(Z<(\mu-124)/(\sigma\sqrt{2})).$$Hence$$\frac{4}{\sigma}=\frac{\mu-124}{\sigma\sqrt{2}}\implies\mu=124+4\sqrt{2}.$$