Exponential Distribution
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Exponential Distribution |
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The exponential distribution is used to model items with a constant failure rate, usually electronics. The exponential distribution is closely related to the poisson distribution. If a random variable, x, is exponentially distributed, then the reciprocal of x, y=1/x follows a poisson distribution. Likewise, if x is poisson distributed, then y=1/x is exponentially distributed. Because of this behavior, the exponential distribution is usually used to model the mean time between occurrences, such as arrivals or failures, and the poisson distribution is used to model occurrences per interval, such as arrivals, failures or defects. The exponential probability density function is
The exponential probability density is also written as
where θ is the mean. From the equations above, it can be seen that λ=1/θ. The variance of the exponential distribution is equal to the mean squared.
The exponential probability density function is shown in the figure below.
The exponential reliability function is
The exponential hazard function is
The exponential hazard function is shown in figure the figure below.
Example: A resistor has a constant failure rate of 0.04 per hour. What is the resistor's reliability at 100 hours? If 100 resistors are tested, how many would be expected to be in a failed state after 25 hours? Solution: The reliability at 100 hours is
The probability of failing before 25 hours is given by the cumulative distribution function which is equal to one minus the reliability function.
The expected number of resistors in a failed state is 100(0.632)=63.2. Click Here to download this solution in Microsoft Excel. The exponential distribution is characterized by its hazard function which is constant. Because of this, the exponential distribution exhibits a lack of memory. That is, the probability of a survival for a time interval, given survival to the beginning of the interval, is dependent ONLY on the length of the interval, and not on the time of the start of the interval. For example, consider an item that has a mean time to fail of 150 hours that is exponentially distributed. The probability of surviving through the interval 0 to 20 hours is
The probability of surviving the interval 100 to 120 is equal to
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