Generating Functions help

Oct 2012
92
0
san francisco
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.
 

chiro

MHF Helper
Sep 2012
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Australia
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.
 
Last edited:
Oct 2012
92
0
san francisco
Nope that is how the question is, so there should be some kind of answer
oh and 7+7+7=21 not 24
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
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.
 
Oct 2012
92
0
san francisco
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
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
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.
 
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Oct 2012
92
0
san francisco
oh alright I get it now thanks for your help