Useful Life The time t until failure of an appliance is...

Useful Life The time $t$ until failure of an appliance is exponentially distributed with a mean of 2 years.
(a) Find the probability density function of the random variable $t$
(b) Find the probability that the appliance will fail in less than 1 year.

Key Concepts

Cumulative Distribution Function (cdf) and Failure Probability

The cumulative distribution function (cdf) of a continuous random variable specifies the probability that the variable takes a value less than or equal to a threshold. For the exponential distribution, the cdf is given by F(t) = 1 ? e^(??t), which is used to compute the probability of failure within a specified time period.

Mean and Rate Parameter Relationship

For an exponential distribution, the mean is the expected value (average time until an event occurs) and is given by 1/?. This relationship allows one to calculate the rate parameter (?) directly from the mean, making it possible to fully define the distribution's pdf.

Exponential Distribution

The exponential distribution is a continuous probability distribution that is commonly used to model the time between events, such as the time until failure of a device. It is characterized by a constant hazard rate (failure rate) and the memoryless property, meaning that the probability of an event in the future is independent of the past.

Probability Density Function (pdf)

The probability density function (pdf) represents the likelihood that a continuous random variable will take on a specific value. In the case of the exponential distribution, the pdf is defined by f(t) = ?e^(??t) for t ? 0, where ? is the rate parameter, which is the reciprocal of the mean.

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