Electronic Device The time to failure of a component in an...

Electronic Device The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without falling for at least five hours. Choose one of the following. Source: Society of Actuaries. $$ \begin{array}{lllllll}{\text { a. } 0.07} & {\text { b. } 0.29} & {\text { c. } 0.38} & {\text { d. } 0.42} & {\text { e. } 0.57}\end{array} $$

Electronic Device The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without falling for at least five hours. Choose one of the following. Source: Society of Actuaries.
$$
\begin{array}{lllllll}{\text { a. } 0.07} & {\text { b. } 0.29} & {\text { c. } 0.38} & {\text { d. } 0.42} & {\text { e. } 0.57}\end{array}
$$

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution commonly used to model the time until an event occurs, such as failure of a component or arrival times in a Poisson process. It is characterized by a constant rate parameter, which implies that the event’s probability of occurring in the next instant is independent of how much time has already elapsed.

Parameter Determination from the Median

In the context of the exponential distribution, the median is the value at which the cumulative probability reaches 0.5. This relationship is used to solve for the rate parameter, allowing one to calibrate the distribution based on known data points such as median lifetime. Determining the rate parameter from the median involves inverting the cumulative distribution function.

Survival Function

The survival function, also known as the reliability function, provides the probability that the time until an event (e.g., failure) exceeds a given time. For the exponential distribution, it is calculated as the exponential of the negative product of the rate parameter and the time threshold, making it a crucial tool for evaluating longevity or reliability of a component.

Probability Calculation using the Exponential Model

Once the rate parameter is determined from a known statistic such as the median, the survival function is used to calculate the probability of survival beyond a specified time frame. This involves substituting the time of interest into the function, thus quantifying the likelihood that the component continues to operate without failure beyond that time.

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