Show from first principles that if X~B(4,0.8), then the probability generating function of X is
Gx(t) = (0.2+0.8t)^4
#pgf
mighty totem
Am I allowed to prove it for the general case
P(X=x) = (nCx)p^x (1-p)^(n-x)
then sub in n = 4 and p = 0.8 at the end, after estabilishing
Gx(t) = (pt+ (1-p))^n
Is that basically the same thing
old temple
Yeah if you can show that sum of nCx p^x (1-p)^(n-x) gives you that expansion
mighty totem
wait but
isnt it by inspection
old temple
It’s probably more work but it does show the result
old temple
wait but
isnt it by inspection
No
mighty totem
like when u do t^x * P(X=x)
old temple
You’ll need to explain how it is by inspection
mighty totem
hm
so like i could say let a = pt or smth
show that its a binomial expansion
basically
old temple
Yeah exactly
old temple
You do need to explain it
But if you explain that it is “obviously” an expansion then it’s fine to sub in n=4 etc
mighty totem
It’s probably more work but it does show the result
I think its acc faster because im not really thinking about any numbers
its js general case which I already know how to prove