What is the purpose of the decay time distribution equation?

[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.

 

[haruspex]

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?

 

[haruspex]

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?

I see no suggestion that this is to do with finding λ. Rather, it assumes you have already determined λ and now wish to estimate the rate of decays in a sample at some future point.