The probability that the component, v$\_($x$),$ of a molecule's velocity is in the interval $v_x\text{'}{v}_x+d{v}_x$ is $f(v_x)d{v}_x$

$f(v_x)=\sqrt[\phantom{2}]{\frac{-m}{2k_{B}T}}{e}^{-\frac{m_x^2}{2}{k}_{B}T}$

a$)$ Show that the probability that ec component v$\_($x$)$ is in the $-$uv$\_($xu is given by$P_{\text{r}\text{o}\text{b}\text{.}}=\frac{2}{\sqrt[\phantom{2}]{}}{\displaystyle \underset{0}{\overset{{}_0}{}}}{e}^{-w^2}dw-=$erf$({}_0)$

$($b$)$ The integral is evaluated numerically. Here are some values: $($look for the values in the picture$)$

Calcalate the fraction of molecules with velocity

i$)v_x\sqrt[\phantom{2}]{\frac{k_{B}T}{m}}$

ii$)v_x\sqrt[\phantom{2}]{\frac{2k_{B}T}{m}}$