You have confused the distribuion of waiting time and the failure rate.

Let f(t) denote the probability density of time T to failure, i.e. the probability of failure in the interval [t, t+dt] is f(t)*dt. Let F denote the distribution function, i.e. F(t)=P(T

Thanks for your answer. Actually, the question should be does failure function represents the probability density function of failure rate? I cannot edit the question right now.

Dear Parag Sen,

your problem concerns the relationship between Poisson distribution and exponential distribution - namely:

If the random variable X represents the number of errors (system failures) in a given time period and has the Poisson distribution, then the intervals between every two consecutive errors have the exponential distribution.

For example, see Wikipedia:

If for every t > 0 the number of arrivals in the time interval [0, t] follows the Poisson distribution with mean value λt, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1/λ.

https://en.wikipedia.org/wiki/Poisson_distribution

In general, the exponential distribution describes the distribution of time intervals between every two subsequent Poisson events.

The answer to your question can be found at the following addresses:

• 7. Reliability (see the the Pages 3 and 4) https://cosmologist.info/teaching/STAT/CHAP7.pdf
• Relation between the Poisson and exponential distributions. https://neurophysics.ucsd.edu/courses/physics_171/exponential.pdf
• Relationship of Poisson and exponential distributions http://www.csee.usf.edu/~kchriste/tools/poisson.pdf
• Exponential Distribution and Poisson Process; see Page 9 http://www.utdallas.edu/~metin/Probability/Folios/expoPois.pdf
• PROBABILITY DISTRIBUTIONS, see the Pages: 8 –20 http://176.32.89.45/~hideaki/res/lecture/ipm/1_Poisson_process.pdf

Best regards

Anatol Badach

Thank you very much, Dr. Anatol Badach, that is the answer I was looking for.