PDF exponential distribution

**kelli_rie** · February 13th 2009, 09:27 AM

Help please!

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.

**mr fantastic** · February 13th 2009, 12:02 PM

Originally Posted by kelli_rie

Help please!

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 $U = \frac{X_1 + X_2}{2}$ . 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).

**CaptainBlack** · February 13th 2009, 02:00 PM

Originally Posted by mr fantastic

You need the pdf for $U = \frac{X_1 + X_2}{2}$ . 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 $t_1$ amd $t_2$ , then the time to failure of the system has an exponential distribution with mtbf $t_3=[t_1^{-1} + t_2^{-1}]^{-1}$ .

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

CB

**kelli_rie** · February 14th 2009, 06:01 PM

Originally Posted by mr fantastic

You need the pdf for $U = \frac{X_1 + X_2}{2}$ . 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!

**matheagle** · February 14th 2009, 07:34 PM

Correct, just use the product of Moment Generating Functions.

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