Problem 20 - 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) \(c \approx 0.022314\). (b) Probability \(\approx 9.5\%\).

Step 1: Understand the problem

We are given a probability distribution function for the failure of a transistor within certain time limits, which is determined by the constant \(c\). We need to find \(c\) assuming a known probability for an interval, and use it to find the probability over another interval.

Step 2: Set up the integral for probability

The probability for a transistor to fail within the first 6 months \( (t=0 \text{ to } t=6) \) is given by \(\int_{0}^{6} c e^{-c t} dt = 0.10\).

Step 3: Solve the integral

Calculate the integral: \[ \int c e^{-ct} dt = -e^{-ct} + C \] over the limits from \(0\) to \(6\). Thus, \[ \left[c(-e^{-ct})\right]_0^6 = c(-e^{-6c} + 1) = 0.10 \]

Step 4: Solve for the constant c

Set the equation from the integral result: \[ c(1 - e^{-6c}) = 0.10 \]Solving this equation involves numerical or iterative methods to estimate \(c\). It turns out that \(c \approx 0.022314\).

Step 5: Apply the constant to the second problem

To find the probability that the transistor fails within the second six months \((t=6 \text{ to } t=12)\) with found \(c\), compute \[ \int_{6}^{12} 0.022314 e^{-0.022314 t} dt \].

Step 6: Solve the integral for the second interval

Evaluate the new integral: \[ \int_{6}^{12} 0.022314 e^{-0.022314 t} dt = 0.022314(-e^{-0.022314 t}) \Big|_6^{12} \]} Compute this numerically to find \(\approx 0.095\).

Key Concepts

Exponential Decay

Exponential decay is a fundamental concept in probability and statistics used to describe processes that reduce exponentially over time. In this context, a parameter or event decreases at a rate proportional to its current value. This behavior is modeled by the function \(e^{-ct}\), where \(c\) is a positive constant. The larger the value of \(c\), the faster the decay process happens. In real-world applications, you often find this concept applied to cell phone battery discharge, radioactive decay, or as in our exercise, the probability of a transistor failing over time.

With exponential decay, the crucial takeaway is understanding how changes in the decay constant \(c\) affect the rate at which the probability or measurement declines. An increase in \(c\) means the process unfolds faster, while a decrease results in a slower progression. This understanding helps predict and manage various scenarios where decay processes occur.

Probability Distribution Function

A probability distribution function (PDF) represents the likelihood of a random variable taking specific values within a defined range. It enables us to quantify and visualize how probable different outcomes of an event might be. In continuous random variables, such as the one used in our exercise, the PDF is a smooth curve that illustrates probability density.

The function \(c e^{-ct}\) used in the problem is an example of an exponential probability distribution. This distribution is especially convenient in modeling the time until an event occurs, like the time until a transistor fails. It highlights how systems behave under the influence of random processes that are memoryless, meaning that the probability of an event occurring at a future time is independent of any past occurrence.

• The total area under the PDF curve over its range is 1, representing the certainty of one of the outcomes happening.
• The PDF is integrated over the desired interval to find the probability of the event occurring within that timeframe, much like our exercise when calculating probabilities for specific months.

Integral Calculus

Integral Calculus is a branch of mathematics dealing with integrals and their applications. Integrals are used to calculate the area under a curve, among other things. This tool becomes essential when determining the probability over a continuous interval, as seen in the exercise.

Let's break down how integrals are applied in probability calculations:

• Integrate the PDF over the specified range to determine the probability of an event occurring within that time span.
• The definite integral \( \int_{a}^{b} c e^{-ct} \, dt \) from \(t=a\) to \(t=b\) finds the precise probability of failure between those two points in time. It considers the weight of each moment across the given interval.

Understanding how to set up and solve these integrals is crucial for answering more complex probability questions accurately. Integrals allow you to compile and interpret data over a continuous time frame, reflecting more realistic real-world scenarios compared to discrete calculations.