# $E \left( \Phi \left( X+ a \sqrt{\frac{V}{n}} \right) \right)=P(T<\frac{a}{\sqrt{1+\sigma^2}})$ where $X\sim N(0 , \sigma^2)$ , $V \sim \chi^2_{n}$

Prove that
$$E\left( \Phi \left(X+ a \sqrt{\frac{V}{n}} \right) \right) =P(T<\frac{a}{\sqrt{1+\sigma^2}})$$ where $$X\sim N(0 , \sigma^2)$$ , $$V \sim \chi^2_{n}$$ is chi-square distribution with $$n$$ degree of freedom , $$X$$ and $$V$$ are independent , $$T$$ has Student's t-distribution with $$n$$ degrees of freedom and $$\Phi$$ is c.d.f of the standard normal distribution, that is $$\Phi(y)=\int_{-\infty}^{y}\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}} \, dt$$.

$$E\left( \Phi \left(X+ a \sqrt{\frac{V}{n}} \right) \right) =\int_0^{+\infty} \int_{-\infty}^{+\infty} \Phi \left(x+ a \sqrt{\frac{v}{n}} \right) f_X(x) f_V(v) dx dv$$

$$=\int_0^{+\infty} \int_{-\infty}^{+\infty} \left(\int_{-\infty}^{ x+ a \sqrt{\frac{v}{n}}} f_Y(y) dy \right) f_X(x) f_V(v) dx dv$$

$$=\iiint_{\{Y

Where $$Y\sim Normal(0,1)$$

$$=P(Y

so $$Y-X\sim N(0,1+\sigma^2)$$ and $$\frac{\frac{Y-X}{\sqrt{1+\sigma^2}}}{\sqrt{\frac{V}{n}}}\sim t_n$$(Student's t-distribution)

$$=P( \frac{\frac{Y-X}{\sqrt{1+\sigma^2}}}{\sqrt{\frac{V}{n}}} <\frac{a}{\sqrt{1+\sigma^2}})$$

$$=P(T<\frac{a}{\sqrt{1+\sigma^2}})$$ where $$T$$ has Student's t-distribution with $$n$$ degrees of freedom.