# 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 **1**st success then the mean is just **1**/p.

**Have more questions or want help passing Exam P?**Â 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!!