Problem 224 - Applied Statistics and Probability for Engineers

Question

An article in Journal of Theoretical Biology ["Computer Model of Growth Cone Behavior and Neuronal Morphogenesis" \((1995,\) Vol. \(174(4),\) pp. \(381-389)]\) developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in the development of the nervous system. This model assumes that the time interval between filopodium formation (a process in growth cone behavior) is exponentially distributed with a mean of 6 time units. Determine the following: (a) Probability formation requires more than nine time units (b) Probability formation occurs within six to seven time units (c) Formation time exceeded with probability 0.9

Step-by-Step Solution

Answer

(a) \( \approx 0.2231 \), (b) \( \approx 0.0561 \), (c) \( \approx 0.6329 \) time units.

Step 1: Understanding the Exponential Distribution

The problem involves an exponential distribution, which is characterized by its rate parameter \( \lambda \). Given that the mean of the distribution is 6 time units, we can find \( \lambda \) using the formula: \( \text{mean} = \frac{1}{\lambda} \), so \( \lambda = \frac{1}{6} \).

Step 2: Calculating Probability for Part (a)

We want to find \( P(X > 9) \) for an exponential distribution. The cumulative distribution function (CDF) of an exponential distribution is \( F(x) = 1 - e^{-\lambda x} \). Therefore, the probability that the time is more than 9 is \( 1 - F(9) = e^{-\frac{9}{6}} = e^{-1.5} \).

Step 3: Solving for the Probability in Part (a)

Evaluating \( e^{-1.5} \approx 0.2231 \), which means \( P(X > 9) \approx 0.2231 \).

Step 4: Calculating Probability for Part (b)

We need to calculate \( P(6 < X < 7) \). This is \( F(7) - F(6) \), where \( F(x) = 1 - e^{-\lambda x} \). Thus, \( P(6 < X < 7) = (1 - e^{-\frac{7}{6}}) - (1 - e^{-\frac{6}{6}}) = e^{-1} - e^{-1.1667} \).

Step 5: Solving for the Probability in Part (b)

Calculating the values, we get \( e^{-1} \approx 0.3679 \) and \( e^{-1.1667} \approx 0.3118 \). Therefore, \( P(6 < X < 7) \approx 0.0561 \).

Step 6: Finding the Time for Part (c)

We want to find the time \( t \) such that \( P(X > t) = 0.9 \). This implies \( e^{-\frac{t}{6}} = 0.9 \), leading to \( -\frac{t}{6} = \ln(0.9) \). Solving for \( t \) gives \( t = -6 \ln(0.9) \).

Step 7: Solving for Time in Part (c)

Calculate \(-6 \ln(0.9) \approx 0.6329\). Therefore, the time is approximately \( t \approx 0.6329 \) time units.

Key Concepts

Probability Calculation

When dealing with exponential distributions, one of the core concepts is determining the probability of certain events occurring within a given timeframe. In the context of neuronal morphogenesis, this involves finding the likelihood that a formation event occurs in a specified interval or a specific time frame.
Probability calculations in exponential distributions are crucial for predicting the behavior of events over time.

• For example, to calculate the probability that an event takes more than a certain number of time units, you use the cumulative distribution function (CDF). This involves computing the probability of the complement event, which involves integrating the probability density function.
• For scenarios where you need to find out probabilities between two time points, like in part (b) of our exercise, you'll determine the difference in the CDF values at those points.

These techniques provide not just answers to these specific problems but also tools to tackle a variety of similar probabilistic scenarios.

Rate Parameter

The rate parameter, or \( \lambda \), is a fundamental part of the exponential distribution. It's intricately tied to the mean of the distribution. In our exercise, the mean is given as 6 time units.
This directly translates into the rate parameter using the formula:

• \( \lambda = \frac{1}{\text{mean}} \)

Thus, for our scenario, \( \lambda = \frac{1}{6} = 0.1667\) (approximately).
The rate parameter influences how "fast" events are expected to occur. A smaller \( \lambda \) means events are less frequent, while a larger \( \lambda \) suggests more frequent occurrence.
Understanding \( \lambda \) allows you to predict the distribution of events over time, which is critical in models like neuronal morphogenesis where timing can indicate biological pathways and behaviors.

Cumulative Distribution Function

The cumulative distribution function (CDF) is an essential tool in the study of random variables. For an exponential distribution, the CDF is defined as:

• \( F(x) = 1 - e^{-\lambda x} \)

The CDF gives the probability that a random variable is less than or equal to a certain value \( x \). This is crucial for understanding not just when an event is likely to occur, but also when it is unlikely.
In problems like our exercise, the CDF helps in determining intervals, such as the probability of formation occurring within 6 to 7 time units.
Additionally, when you want to calculate the probability of events taking more than certain time units, you utilize the complement of the CDF's output, providing a quick way to solve probability questions effectively.

Neuronal Morphogenesis

Neuronal morphogenesis refers to the development and growth of neurons, a vital process in forming the nervous system. This process involves various elements, including growth cones.
A growth cone is a dynamic, motile structure present at the tip of a developing axon, responsible for sensing the environment to guide the axon to its destination.
Understanding the timing and dynamics of growth cone behavior is crucial, as it influences

• how neurons connect and form networks
• the overall architecture of the nervous system

The exponential distribution model, like the one mentioned in the exercise, is used to predict and characterize the behavior of these growth cones.
This statistical approach provides insights into biological processes, aiding in the formation of hypotheses and the development of further research in neurobiology.