# PDF exponential distribution

• Feb 13th 2009, 09:27 AM
kelli_rie
PDF exponential distribution

Suppose that two electronic components in the guidance system for a missile operate independently and that each has a length of life governed by the exponential distribution with mean 1 (with measurements in hundreds of hours).
Find the probability density function for the average length of life of the two components.
• Feb 13th 2009, 12:02 PM
mr fantastic
Quote:

Originally Posted by kelli_rie

Suppose that two electronic components in the guidance system for a missile operate independently and that each has a length of life governed by the exponential distribution with mean 1 (with measurements in hundreds of hours).
Find the probability density function for the average length of life of the two components.

You need the pdf for $\displaystyle U = \frac{X_1 + X_2}{2}$. I'd just use a moment generating function approach to get this.

Note: The pdf for the sum of i.i.d. exponential distributed random variables is the Erlang distribution (which is a special case of the gamma distribution).
• Feb 13th 2009, 02:00 PM
CaptainBlack
Quote:

Originally Posted by mr fantastic
You need the pdf for $\displaystyle U = \frac{X_1 + X_2}{2}$. I'd just use a moment generating function approach to get this.

Note: The pdf for the sum of i.i.d. exponential distributed random variables is the Erlang distribution (which is a special case of the gamma distribution).

I think this is probably a misinterpretation of the question.

If we are interested in the distribution of the time to failure of the system, which has two failure modes with time to failure distributed exponentialy with mtbf $\displaystyle t_1$ amd $\displaystyle t_2$, then the time to failure of the system has an exponential distribution with mtbf $\displaystyle t_3=[t_1^{-1} + t_2^{-1}]^{-1}$.

That is the failure rate for the system is the sum of the failure rates of the subsystems.

CB
• Feb 14th 2009, 06:01 PM
kelli_rie
Quote:

Originally Posted by mr fantastic
You need the pdf for $\displaystyle U = \frac{X_1 + X_2}{2}$. I'd just use a moment generating function approach to get this.

Note: The pdf for the sum of i.i.d. exponential distributed random variables is the Erlang distribution (which is a special case of the gamma distribution).

Thank you so much. It seems so obvious now that you say it!
• Feb 14th 2009, 07:34 PM
matheagle
Correct, just use the product of Moment Generating Functions.