Why does small change in Base current causes a larger change in collector current in case of transistors? Why does small change in the base current causes larger change in the collector current. That is say if I increase the base current buy a small amount why the collector current increases by a very large amount. Please explain.
 A: this is a hand-wavy kind of answer that will probably get down-voted.
the collector-base junction is normally reverse-biased in a circuit where a BJT is used as a current amplifier.  because it's reverse-biased the material on both sides of the junction are then depleted of charge carriers for the same reason a reverse-biased diode is.  then, without charge carriers (holes or electrons), the semiconductor material acts as an insulator.  pretty much like glass.
now the base is a very thin layer that exists between the collector and emitter and doped opposite of the collector and emitter.  if the base-emitter junction is forward-biased there will be charge carriers moving across the junction.  once these charge carriers get across the base-emitter junction (electrons in one direction and holes in the other), they will begin to recombine (electron-hole recombination) but that takes time and, before they recombine, they remain charge carriers in a current.  now a large fraction of those charge carriers will not yet recombine and will be drawn across the thin base (due to a high $V_\text{cb}$ voltage) and get to the collector-base junction which would otherwise be depleted of charge carriers.  only a small fraction will be recombined and be drawn off in the base terminal connection.
the large fraction of charge carriers remaining will be looking at a large and juicy $V_\text{cb}$ potential that will draw them across the collector-base junction to the collector terminal connection.
but the fraction remains relatively constant.  so if a small percentage of the emitter current is the base terminal current and a much larger percentage is the collector current then a small change in the base current must cause a larger change in the collector current.
this is a qualitative explanation.  to do it right, one would have to derive the Eber-Molls volt-amp equations for the BJT.  being that i haven't done it for 40 years, i will pass on that, but i can show you how to use the Eber-Molls equations (along with typical parameters), reverse bias the collector-base junction with a large voltage, and derive current gain from that.
