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In the study of transistor as an amplifier, if $\alpha=I_c / I_e$ and $\beta=I_c / I_b$, where $I_c, I_b$ and $I_c$ are the collector, base and emitter currents, then
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The correct answer is:
$\beta=\frac{\alpha}{1-\alpha}$
As we know $i_E=i_C+i_B$
$\Rightarrow \frac{i_e}{i_c}=1+\frac{i_b}{i_c} \Rightarrow \frac{1}{\alpha}=1+\frac{1}{\beta} \Rightarrow \beta=\frac{\alpha}{1-\alpha}$
$\Rightarrow \frac{i_e}{i_c}=1+\frac{i_b}{i_c} \Rightarrow \frac{1}{\alpha}=1+\frac{1}{\beta} \Rightarrow \beta=\frac{\alpha}{1-\alpha}$
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