For a bipolar junction transistor to operate properly, the base-emitter junction must be forward biased while the collector-base junction must be reverse biased. This will place $$V_{BE}$$ at approximately 0.7 volts and the collector current $$I_C$$ will be equal to the base current $$I_B$$ times the current gain $$beta$$. For small signal devices, the current gain is greater than 100 typically. Thus, $$I_C \gg I_B$$ and $$I_C \approx I_E$$.
The common emitter configuration places the emitter terminal at ground. The base terminal is seen as the input and the collector as the output. Using a fixed base supply, the base current is dependent on the value of the base resistor via Ohm’s law. Consequently, any variation in current gain across a batch of transistors will show up as an equivalent variation in collector current, and by extension, a variation in collector-emitter voltage $$V_{CE}$$.