Alpha/Beta Parameters of BJT
The first equation you wrote is correct.

$$I_E = I_C+I_B \tag1 $$ 

But he other two equations are written by neglecting the reverse saturation currents \$I_{CBO}\$ and \$I_{CEO}\$. The original equations are: 

$$\begin{align} I_C &amp;= \alpha I_E + I_{CBO} \tag{2}\\
I_C &amp;= \beta I_B + I_{CEO}\\
&amp;=\beta I_B + (\beta +1)I_{CBO} \tag{3}\end{align}$$