Generating Functions help

Hey there everyone, I need some help with this math problem, hoping to get some help on this topic.

Use generating functions to find the number of solutions in integers to the equation a+b+c=30 where each variable is at least 3 and at most 7.

Please explain to me step by step on how to do this so I can understand it better thanks.

Re: Generating Functions help

Hey gfbrd.

Given your constraints you will never get a solution since your maximum value will be 7+7+7=21 and your minimum will be 3+3+3=9. Did you mean to say something else?

Edit: Can't add up properly.

Re: Generating Functions help

Nope that is how the question is, so there should be some kind of answer

oh and 7+7+7=21 not 24

Re: Generating Functions help

Thanks for pointing out the error.

Well it sounds like what they want you to do is have two independent random variables with 3 to 7 and then have another random variable which is 30 minus the sum of those two.

The sum of two random variables' distribution can be found with a PGF and this will be based on two uniform distributions of 5 values with same probability for 3 to 7 inclusive.

Then the other variable will be 30 - (X+Y) but 30 is just a special case of a distribution where you have Z - W where Z has a probability density function of P(Z=30) = 1 and the distribution of -W just reflects the distribution around the y-axis.

So can you calculate for a start the PGF for the sum of two uniform random variables (discrete uniform) with values going from 3 to 7 inclusive.

Re: Generating Functions help

lol sorry im not understanding this :(

if you dont mind i hope you can make it more simple for me to understand hahaha sorry im slow at this kind of stuff

Re: Generating Functions help

Let X = Uniform(3,7) and Y = Uniform(3,7) as well. Use the PGF formula to find the probability generating function for X+Y if X,Y are independent.

Re: Generating Functions help

oh alright I get it now thanks for your help