# The product of two normally distributed probabilities [closed]

(Please forgive my layperson's level of knowledge.)

# Our Hypothetical Problem

A small percentage of people infected with Cooties become obsessed with statistical analysis. The Cootie Monster has escaped from his prison island and will infect some of the population before he is captured. What is the likely range of people who will become obsessed with statistical analysis due to the Escape from Cootie Island scandal?

# Translating Into Mathlish

The at-risk population is 7,773,200,000 people.

40–70% of the population will be infected with Cooties. This is a normal distribution and the range is ± 2 standard deviations, therefore: $$\mu_1 = 0.55$$ $$\sigma_1 = 0.075$$ $$\sigma_1^2 = 0.005625$$ (1. Right?)

Of those who are infected with Cooties, multiple things affect the probability of developing the lifelong-affliction of obsession with statistical analysis. The probability across the population is a normal distribution of 0.6–5.9%, therefore: $$\mu_2 = 0.0325$$ $$\sigma_2 = 0.01325$$ $$\sigma_2^2 = 0.0001755625$$ (2. Right?)

# A Pair of Potential Procedures Producing the Product of Probability Progressions (distributions)

John D. Cook, PhD wrote a blog post (if the link is broken, here is a copy at the Internet Archive Wayback Machine) explaining two formulae that might be the method for answering our hypothetical. (I lack the knowledge to completely understand the short post.)

The two formulae are $$\mu = \cfrac {\sigma_1^{-2}\mu_1 + \sigma_2^{-2}\mu_2}{\sigma_1^{-2} + \sigma_2^{-2}}$$ $$\sigma^2 = \cfrac {\sigma_1^2\sigma_2^2}{\sigma_1^2 + \sigma_2^2}$$

# Do the Math!

μ = 0.04816289368

σ² = 0.00017024884

σ ≈ 0.013047944

(3. Right?)

μ * 7,773,200,000 = 374,379,805

σ * 7,773,200,000 = 101,424,278

(n.b. I didn't even attempt to use significant digits at any point (no pun intended), and I rounded μ and σ to the nearest whole person because, uh, Benghazi!)

# Interpreting the Math

In his blog post, John D. Cook, PhD wrote the following immediately after the two formulae.

Note that the product of two normal random variables is not normal, but the product of their PDFs is proportional to the PDF of another normal.

After reading that, I realized that abnormal people are proportional to normal people, but we focus so much on the differences that we don't think see we are mostly the same. That's certainly not what Dr. John was explaining, but I don't understand what he was explaining and this insight about the human condition was my consolation prize.

I'm pretty sure that the product is not normal, but if it's not normal, I don't know how it is distributed.

With my limited understanding, I would express the hypothetical answer as

171,531,249–577,228,361

But that's an ugly range, so maybe something like

171,500,000–577,200,000

And I would imagine a bell curve with 171,500,000 and 577,200,000 being two standard deviations from the mean.

# I'm Wrong, But Where Are My Errors?

At the very least, I'm not interpreting the results properly. He wrote, "that the product . . . is not normal," so because I'm interpreting the resulting μ and σ² as a normal distribution, I must be wrong. (I mean, I guess it's wrong, but my brain hurts because I've been trying to figure out this problem for 12 days. I'm not saying I'm obsessed with statistical analysis, though.)

Am I applying the right formulae to this problem? If these probabilities were not ranges, it would be easy. For example, 7,773,200,000 people * 41.88% infected with Cooties * 3.07% of the infected become obsessed with statistical analysis = 105,429,932.632 people. (If that's not right, I will cry.) So, I feel like I should multiply the two probability distributions to get the right answer.

In statistics, I'm a layperson. In law, I'm not. When most laypeople talk to me about law and then ask a question (by "most", I mean "all"), their understanding of law is so flawed that their question is nonsensical, invalid, or irrelevant. For my sake—it took me three hours to write this question on my phone—and for your sake—I don't want you to slap your forehead—I hope that my question is intelligible. (And, O. M. G., if this is a duplicate question, I will curl up in a ball and cry myself to sleep. I've spent at least 10 total hours searching and reading various websites for the answer.)

One final thing. I want to be wrong. I know that seems contradictory, but I hope I understand the basics of the problem and that I applied the wrong formulae or made an arithmetic error or something. This answer, 171,500,000–577,200,000, doesn't fit with my other calculations using different methods. I was expecting 139,917,600 to be near the mean, but if what I wrote above is correct, then its z-score is -2.31, which is the 1.04th-percentile. I'll stop now: I'll talk to a therapist about the dissonance of my calculations.