The probability of a transistor failing between t=a months...

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?

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?

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Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution commonly used to model the time until an event occurs, such as the failure of a component. It is characterized by a constant failure rate regardless of how much time has already passed, making it ideal for reliability analysis.

Cumulative Distribution Function (CDF)

For the exponential distribution, the CDF gives the probability that the event (e.g., a failure) has occurred by a certain time t, and is expressed as 1 - e^(-ct) for a rate parameter c. This function is essential for calculating the likelihood of failures within specified time intervals.

Parameter Estimation

Parameter estimation involves solving for the rate parameter c when provided with a known probability over a certain period. By setting the CDF equal to the given probability and solving the resulting equation, one obtains the value of c that characterizes the failure rate.

Memoryless Property

A key property of the exponential distribution is its memoryless nature, meaning that the probability of failure in the next period is independent of how much time has already elapsed. This simplifies the calculation of failure probabilities over sequential time intervals, as each interval can be treated independently.

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