B How to combine 2 distributions with different sample sizes?

AI Thread Summary
To combine two distributions with different sample sizes, one effective method is to use weighted averages based on the sample sizes of each distribution. The resulting distribution can be represented as a mixture distribution, which accounts for the probabilities and outcomes of both original distributions. This approach ensures that the contributions of each distribution are proportional to their respective sample sizes. The discussion highlights the importance of properly weighting the distributions to achieve a meaningful combination. Overall, creating a third distribution through this method is feasible and mathematically sound.
batmantrippin
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I would like to know how to combine 2 distributions with different sample sizes and have a new 3rd distribution.
I apologise in advance for what is a very basic question for someone with a maths degree (it was a long time ago!).

I have 2 distributions that look something like this (but with much bigger samples), in the form of (probability,outcome). The outcome is literally just a number.

Distribution 1:
(0.1 , -1)
(0.2 , -0.9)
(0.25 , 0)
(0.3 , 4.5)
(0.15 , 7)

Distribution 2:

(0.05 , -1)
(0.05 , -0.8)
(0.05 , -0.5)
(0.05 , -0.2)
(0.1 , 0)
(0.1 , 3)
(0.1 , 6)
(0.1 , 6.5)
(0.2 , 7)
(0.2 , 7.5)

What is the best way to combine (average?) the two and have a third distribution in the same format that is basically an average of the two? Or am I asking for something that's not really doable?
 
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Add them with weights proportional to the sample sizes.
 
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The procedure described by @mathman gives what is called a mixture distribution.
 
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