Problem 14 - Applied Calculus

Question

The probability of a transistor failing between \(t=a\) months and \(t=b\) months is given by \(c \int_{a}^{b} e^{-c t} d t\), for some constant \(c\). (a) If the probability of failure within the first six months is \(10 \%\), what is \(c\) ? (b) Given the value of \(c\) in part (a), what is the probability the transistor fails within the second six months?

Step-by-Step Solution

Answer

(a) \(c = -\frac{\ln(0.90)}{6}\). (b) Probability in second six months is \(e^{-6c} - e^{-12c}\).

Step 1: Understanding the Problem

We need to find the constant \(c\) such that the probability of a transistor failing within the first six months is 10%. We will use this to find the probability of failure in the second six months.

Step 2: Calculating c for the First Interval

Given \(c \int_{0}^{6} e^{-ct} dt = 0.10\), we need to solve for \(c\). Start by evaluating the integral: \(\int e^{-ct} dt = -\frac{1}{c} e^{-ct} + C\). Plug in the limits: \[\int_{0}^{6} e^{-ct} dt = \left[-\frac{1}{c} e^{-ct}\right]_{0}^{6} = -\frac{1}{c} e^{-6c} + \frac{1}{c} e^{0}= \frac{1}{c} - \frac{1}{c} e^{-6c}.\] Set the equation \(c\left(\frac{1}{c} - \frac{1}{c} e^{-6c}\right) = 0.10\) and solve for \(c\). This simplifies to \(1 - e^{-6c} = 0.10\) or \(e^{-6c} = 0.90\). Take the natural logarithm: \(-6c = \ln(0.90)\). Thus, \(c = -\frac{\ln(0.90)}{6}\).

Step 3: Determining the Probability of Failure in the Second 6 Months

With \(c\) determined, calculate \(c \int_{6}^{12} e^{-ct} dt\). This integral is \[\int_{6}^{12} e^{-ct} dt = \left[-\frac{1}{c} e^{-ct}\right]_{6}^{12} = -\frac{1}{c} e^{-12c} + \frac{1}{c} e^{-6c}.\] Simplify and then plug the known \(c\) value. Thus, the probability of failure in the second six months is \(c\left(\frac{1}{c}e^{-6c} - \frac{1}{c}e^{-12c} \right)\), which becomes \(e^{-6c} - e^{-12c} \).

Key Concepts

Exponential Distribution

In probability theory, an exponential distribution is useful for modeling the time until an event happens, such as the failure of a transistor. This distribution is characterized by a constant rate of decay, depicted by the formula \( e^{-ct} \), where \( c \) is a positive constant, and \( t \) is the time interval.

This distribution finds applications in various fields, such as reliability engineering, where it models the lifespan of electronic components. A key feature is its memoryless property, where the probability of failure in future times does not depend on how much time has already elapsed.

Understanding exponential distribution allows one to calculate the probability of events occurring over time intervals, which is central to problems like determining the likelihood of a transistor failing.

Integration Techniques

Integration is a mathematical operation used to determine accumulation, such as area under a curve. When dealing with an exponential distribution, integration helps calculate probabilities over specific intervals of time.

The integral of \( e^{-ct} \) with respect to \( t \) is fundamentally important. This anti-derivative is calculated as \(-\frac{1}{c} e^{-ct} + C\), with \( C \) being the constant of integration. For practical probability calculations, we evaluate definite integrals, like \( \int_{a}^{b} e^{-ct} dt \), by finding the difference between its values at bounds \( b \) and \( a \).

Integration techniques are therefore vital in solving problems with exponential distributions, enabling the computation of failure probabilities over specified periods.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function representing the inverse of the exponential function. In problems dealing with exponential decay, the natural logarithm is key to solving for variables like the constant \( c \).

For instance, converting an equation like \( e^{-6c} = 0.90 \) into \( -6c = \ln(0.90) \) employs the natural logarithm, facilitating the isolation and calculation of \( c \).

Understanding the natural logarithm simplifies equations derived from exponential functions, making it indispensable for solving probability-related equations.

Probability Calculation

Probability calculations provide insights into how often an event may occur within a certain timeframe. With transistors, calculating the probability of failure between two points in time involves understanding and applying the exponential distribution and integration principles.

The general formula \( c \int_{a}^{b} e^{-ct} dt \) determines the probability between times \( t = a \) and \( t = b \). Solving this with known or calculated values, like the constant \( c \), results in a numerical representation of failure likelihood.

Performing these computations accurately ensures a clearer view of the reliability of systems and components over time, helping predict their performance and manage operational risks efficiently.