Question

The probability for a radioactive particle to decay between time \(t\) and time \(t+d t\) is proportional to \(e^{-\lambda t} .\) Find the density function \(f(t)\) and the cumulative distribution function \(F(t) .\) Find the expected lifetime (called the mean life) of the radioactive particle. Compare the mean life and the so-called "half life" which is defined as the value of \(t\) when \(e^{-\lambda t}=1 / 2\).

Short answer

The density function is \( f(t) = \lambda e^{-\lambda t} \). The cumulative distribution function is \( F(t) = 1 - e^{-\lambda t} \). The mean lifetime is \( \frac{1}{\lambda} \) and the half-life is \( \frac{\ln(2)}{\lambda} \).

Step-by-step solution

- Identify the given information

The probability of decay between time \( t \) and \( t + dt \) is proportional to \( e^{-\lambda t} \). This suggests an exponential distribution.

- Set up the density function

The probability density function (pdf) for an exponential distribution is \( f(t) = \lambda e^{-\lambda t} \), where \( \lambda \) is the rate parameter. The function is derived from the given condition that probability is proportional to \( e^{-\lambda t} \).

- Find the cumulative distribution function

The cumulative distribution function (CDF) is \( F(t) = \int_0^t f(s) \, ds \). For the exponential distribution, we calculate: \[ F(t) = \int_0^t \lambda e^{-\lambda s} \, ds = 1 - e^{-\lambda t} \]

- Find the expected lifetime

The expected lifetime, or mean life \( E(T) \) for an exponential distribution is \( \frac{1}{\lambda} \). This is derived from the mean of the exponential distribution.

- Calculate the half-life

The half-life \( t_{1/2} \) is defined as the time when \( e^{-\lambda t} = \frac{1}{2} \). Solving for \( t \):\[ e^{-\lambda t_{1/2}} = \frac{1}{2} \]Taking the natural logarithm of both sides,\[ -\lambda t_{1/2} = \ln \left( \frac{1}{2} \right) \]\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]

- Compare the mean life and half-life

Compare \( E(T) = \frac{1}{\lambda} \) and \( t_{1/2} = \frac{\ln(2)}{\lambda} \). The half-life is shorter than the mean life because \( \ln(2) < 1 \).

Key concepts

exponential distribution

When dealing with the probability of a radioactive particle decaying over time, we often use the exponential distribution. This distribution is crucial when the likelihood of an event happening is proportional to a decreasing exponential function.
In our case, the probability of decay between time \(t\) and time \(t + dt\) is given by an exponential function. This helps us model scenarios where the event rate remains constant over time, which is common in radioactive decay processes.

probability density function

To describe how the decay process behaves over time, we use the probability density function (pdf). In the context of an exponential distribution, the pdf is represented by:
\[f(t) = \lambda e^{-\lambda t}\]
Here, \( \lambda \) is the rate parameter, which determines how quickly the radioactive particles decay. The pdf helps us understand the instantaneous rate at which the particles decay at any given time \(t\). It tells us that the probability of decay decreases exponentially as time progresses.

cumulative distribution function

Next in our understanding is the cumulative distribution function (CDF), which gives us the probability that a particle will decay by time \(t\). For the exponential distribution, the CDF is calculated as:
\[F(t) = \int_0^t \lambda e^{-\lambda s} \, ds = 1 - e^{-\lambda t}\]
This function accumulates the probabilities over time. Unlike the pdf, which focuses on the instantaneous rate, the CDF considers the total probability up to a certain time, providing a more comprehensive picture of the decay process.

mean life

The mean life, also known as the expected lifetime, is a key statistic in our analysis. For an exponential distribution, the mean life \(E(T)\) is calculated as:
\[E(T) = \frac{1}{\lambda}\]
This average value helps us predict how long, on average, a radioactive particle will exist before decaying. The mean life is inversely proportional to the rate parameter \(\lambda\), meaning the higher the decay rate, the shorter the mean life.

half-life

Another important concept is the half-life, which is the time required for half of the radioactive particles to decay. It’s defined by the equation:
\[e^{-\lambda t_{1/2}} = \frac{1}{2}\]
Solving for \(t_{1/2}\), we get:
\[t_{1/2} = \frac{\ln(2)}{\lambda}\]
Understanding the half-life helps in practical applications, such as determining how long it takes for a substance to lose half of its activity.

rate parameter

The rate parameter \(\lambda\) is a fundamental aspect of the exponential distribution. It determines how quickly the events (decays) occur. A higher \(\lambda\) indicates a faster decay process, while a lower \(\lambda\) suggests a slower process.
It’s essential to note that both the mean life and half-life are directly influenced by \(\lambda\):

• Mean life: \(E(T) = \frac{1}{\lambda}\)
• Half-life: \(t_{1/2} = \frac{\ln(2)}{\lambda}\)

By understanding \(\lambda\), we can gain insights into the pace at which radioactive particles decay.