Nov
09
Find the moment-generating function of the discrete random varible X that has the probability distribution..?
By
f(x) = 2(1/3)^x for x = 1,2,3…And use it to determine the values of u`(subscript 1) and u`(subscript 2)
I understand the definition of moment generating function, but how do we do this problem since x goes to infinity?
My teacher gave me the solution, but he skipped a lot of steps so I didn’t understand how he got what he did
So please, show your steps and a little explanation too
Much appreciated!
1 Comments
November 9th, 2009 at 7:02 pm
You just need to realize that it is defined as:
M(t) = E( e^(tX) ) = ∑ e^( t x[i] ) f( x[i] )
where X is your random variable. Then you just put in what you know:
M(t) = ∑ e^( t x[i] ) 2 (1/3)^x[i]
Since we are doing x=1,2,3… I will write this as a sum instead:
M(t) = 2 ∑ e^( n t ) (1/3)^n
M(t) = 2e^t ∑ (e/3)^n
Now, that’s a geometric series, but it’s n=1 to ∞ (not n=0), so:
M(t) = 2e^t ( 1 / (1-e/3) – 1 )
Done.