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# Two different geometric mean formulas… which is right?

If you’ve been studying for Exam P and have come across two different formulas for the mean of a geometric distribution, you’re not alone! It’s a question I get asked a lot, particularly when members of my Study Strategy Program get to Question #3 of the TIA free practice Exam #4.

Here’s why…

The geometric distribution can be used to define two different types of random variables.

Sometimes, the question states that the number of FAILURES until the first success is geometric. Or, instead, the question may state that the range of the variable is 0,1,2,3, etc.. (take note of the 0).

If that’s the case, the mean formula is (1-p)/p.

Other times, the question states that the number of trials until the FIRST SUCCESS is geometric or instead it might state that the range of the variable is 1,2,3, etc.. (take note of NO 0 in this range, because the “first success” must occur on at least the 1st trial – it cannot occur before that)

In this case the mean formula is 1/p.

If you’re using the ACTEX study manual, this is stated around page 184 of your study manual under “Geometric distribution with parameter p”.

So in TIA Exam #4, question 3, how does the instructor know to use the second formula above?

He looks at the RANGE of the X and Y variables given in the question. The range is one 1,2,3,…. Therefore, he concludes that this geometric distribution is referring to the FIRST SUCCESS and therefore uses the mean formula 1/p.

Now here’s a quick little memorization trick for how to remember the difference between the two.

If the range starts with 1 then the random variable is referring to the 1st success then the mean is just 1/p.

Have more questions or want help passing Exam P?  Join the Etched Actuarial Study Strategy Program for step-by-step guidance on exactly how to pass Exam P + get answers to all your questions along the way!!

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