# Muon decay time distribution

tryingtolearn1
Homework Statement:
Muons decay time distribution
Relevant Equations:
##N(t) = N_0 exp(−\lambda t)## and ##D(t) = \lambda \exp(−\lambda t)##
I know for muons that the the probability that a muon decays in some small time interval ##dt## is ##\lambda dt##, where ##\lambda## is a decay rate. Thus the change in the population of muons is just ##dN/N(t) = −\lambda dt##. Integrating gives ##N(t) = N_0 \exp(−\lambda t)##. This makes sense to me but my book goes on to say the following,

By decay time distribution D(t), we mean that the time-dependent probability that a muon decays in the time interval between ##t## and ##t + dt## is given by ##D(t)dt##. If we had started with ##N_0## muons, then the fraction ##−dN/N_0## that would on average decay in the time interval between ##t## and ##t + dt## is just given by differentiating the above relation: ##−dN = N_0\lambda \exp(−\lambda t) dt## ##\therefore## ##−dN/ N_0 = \lambda \exp(−\lambda t) dt##. The left-hand side of the last equation is nothing more than the decay probability, so ##D(t) = \lambda \exp(−\lambda t)##.

What exactly is that explaining? Don't we need to know what ##\lambda## is before using the ##D(t)## equation? Because trying to find ##\lambda## using ##D(t) = \lambda \exp(−\lambda t)## will give the wrong results.

## Answers and Replies

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Homework Statement:: Muons decay time distribution
Relevant Equations:: ##N(t) = N_0 exp(−\lambda t)## and ##D(t) = \lambda \exp(−\lambda t)##

I know for muons that the the probability that a muon decays in some small time interval ##dt## is ##\lambda dt##, where ##\lambda## is a decay rate. Thus the change in the population of muons is just ##dN/N(t) = −\lambda dt##. Integrating gives ##N(t) = N_0 \exp(−\lambda t)##. This makes sense to me but my book goes on to say the following,

What exactly is that explaining? Don't we need to know what ##\lambda## is before using the ##D(t)## equation? Because trying to find ##\lambda## using ##D(t) = \lambda \exp(−\lambda t)## will give the wrong results.
I'm not entirely sure what you are asking, but it looks to me that D(t) is defined as ##\frac{P(decay in interval (t,t+dt))}{dt}##, whereas the ##\lambda dt## expression assumes it has not decayed at time t.
So D(t)=P(undecayed_at_time (t))λ = ##\lambda \exp(−\lambda t)##

tryingtolearn1
I'm not entirely sure what you are asking, but it looks to me that D(t) is defined as ##\frac{P(decay in interval (t,t+dt))}{dt}##, whereas the ##\lambda dt## expression assumes it has not decayed at time t.
So D(t)=P(undecayed_at_time (t))λ = ##\lambda \exp(−\lambda t)##
Hmm but why would that equation be relevant? Suppose you know what ##t## is and you're trying to find ##\lambda##, why would ##D(t)=\lambda\exp(-\lambda t)## be relevant?