# Finding the product of multiple normal distributions

Charij
Hi all,
I've been working on a little side project, but I've hit a road block on the maths for this one. Basically if you imagine a sealed pack of 10 cards, there is a 20% chance that the pack contains one foil (more valuable) card. The mass distribution of the foil cards are (heavier and) different to the non-foil equivelant.

What I would like to do, is calculate the probability of a sealed pack containing a foil card, given the mass of the pack. Essentially I have 3 normal mass distributions for the non-foil cards, foil cards, and packaging. The next step is to combine the 3 normal distributions to give me a bell-curve that shows the probability of a pack containing a foil, given a mass.

I'm guessing I would need to find the product of the packaging, 9 non-foil cards, and 1 foil card distribution; I've struggled to find information on how to do this. Any guidance, or thoughts on this would be most appreciated!

Thanks,
Charij

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So m=m1+m2+m3 where m1 is the mass of the foil card, m2 of the nine non-foil cards and m3 of the packaging.
The pdf for m is then $p(m)=\int dm_1 \int dm_2 \int dm_3 p_1(m_1) p_2(m_2) p_3(m_3) \delta(m-m_1+m_2+m_3)$
Then use $\delta(x)=1/2\pi \int_{-\infty}^{\infty} dp \exp(ipx)$.
Interchange the order of integration over p and m_1 to m_3.

Charij
Thanks DrDu,
After reading some wiki, you have put me on the right track! It'll take me a little time to work exactly what you're saying, but it seems to make some sense. Any chance you could link me some material on how to do this?
Then use δ(x)=1/2π∫∞−∞dpexp(ipx).
Interchange the order of integration over p and m_1 to m_3.

Thanks again,
Charij

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I'm guessing I would need to find the product of the packaging, 9 non-foil cards, and 1 foil card distributionj

You want the distribution of the sum of those things. The sum of independent normally distributed random variables is a normal random variable, so what you can look up on the web is "sum of independent normal (or 'Gaussian' l random variables". Of course, you can also calculate the answer "from first principles" using Dr DuDu's method.

That distribution doesn't give your final answer because you want to calculate the conditional probability that a pack of a given weight contains a foil card. You have to use Bayes rule to do that.

Charij
Thanks Stephen, that was incredibly useful too! From your information, I plan to use the sum of the normal distributions using convolution. Then use Bayes' Theorem to calculate the probability.