How do you find the Probability Generating Function?

[Orphen89]

Homework Statement

Consider a branching process with branching probabilities given by P₀=1/2 and P_j=1/3$$^{j}$$ for j $$\geq$$ 1

Find the probability generating function: $$\sum^{\infty}_{n=0} p_{n}x^{n}$$

The Attempt at a Solution

Now, the answer is supposed to be G(x) = (x+3)/(2(3-x)), but I really have no idea how to get this answer. I've checked it up on the internet and haven't found anything that looks even vaguely helpful, and since I'm only meeting my tutor later in the week I'm pretty much stuck on my own for this one. Is there some sort of formula you can use to find the probability generating function? If possible, could you please show me how to get that answer from the given data?


Thanks in advance.

 

[Random Variable]

$$G(x) = \frac {1}{2} + \frac {1}{3} x + \frac {1}{9} x^{2} + \frac {1}{27} x^{3} + ...$$

$$1 + \frac {1}{3} x + \frac {1}{9} x^{2} + \frac {1}{27} x^{3} + ... = \frac {1}{1 - \frac {x}{3}} = \frac {3}{3 - x }$$

$$G(x) = \frac {3} {3-x} - \frac {1}{2} = \frac {x+3}{2(3-x)}$$

 

[Orphen89]

Thanks for the quick reply! I just have two questions: In the second line, is it supposed to be 1/2 or 1? And in the third line, you're basically just subtracting the p₀ value from G(x) right?

Thanks for the help.

 

[Random Variable]

Thanks for the quick reply! I just have two questions: In the second line, is it supposed to be 1/2 or 1? And in the third line, you're basically just subtracting the p₀ value from G(x) right?

Thanks for the help.

The second line is a geometric series. But it's 1/2 more than what we want.

 

[Orphen89]

Sorry I just have one more question - does this mean that for p₀= 1/3, p₁=1/4, p₂=1/6 and p_n=1/2^n that the probability generating function is equal to -(7x-6)/(4(x-2)) + 1/3 + (1/4)x + (1/6)x^2?

Also, if it is incorrect, could someone please show me the correct method? For some reason I'm having a fair bit of difficulty getting my mind around this...

Thanks in advance.

 

[Random Variable]

Are you saying that $$p_{n}= (\frac {1}{2})^{n}$$ for n > 2?

Then your answer is correct if $$\sum^{\infty}_{3}p_{n}x^{n} = -\frac {7x-6}{4(x-2)}$$

but that's not what I'm getting

 

[Orphen89]

Sorry about that! I forgot about that part - it is p_n=1/2^n for n $$\geq$$ 3. Thanks for all the help Random Variable, I appreciate it (all this Generating Function stuff is going to be on an exam coming up in a few weeks...)

 

[Random Variable]

I'm getting $$\frac {x^{3}}{4(2-x)}$$

 

[Orphen89]

Really? I think you might be right, because I can remember hearing that that was the answer...

I originally thought that (using the data), I could create a probability generating function with a/(1-a) - 1/3 - 1/4 - 1/6, with a = x/2

Do you know where I have gone wrong, (or if I even have the right equation...?) >_<

 

[Random Variable]

see below

 

[Random Variable]

We know $$1 + \frac {1}{2} x + \frac {1}{4} x^{2} + \frac {1}{8} x^{3} + \frac {1}{16}x^{4} + ... = \frac {1}{1 - \frac {x}{2}}$$ (infinite geometric series with common ratio x/2 and first term of 1)

What we want is $$\frac {1}{8} x^{3} + \frac {1}{16} x^{4} + \frac {1}{32}x^5+ ...$$

so subtract the terms we don't want

$$\frac {1}{1 - \frac {x}{2}} - 1 - \frac {1}{2}x - \frac {1}{4}x^2$$

 

[Random Variable]

As a side note, to find the mean of a discrete probability distribution, take the derivative of the associated probability generating function and evaluate it at x=1.

 

[Orphen89]

Wow, I think I finally understand how to do it now. Thanks for all the help Random Variable, you managed to explain the probability generating function to me better than my tutor =D

 

[Random Variable]

And if you take the second derivative of a probability generating function and evaluate it at x=1, you'll get

$$E(X(X-1)) = E(X^{2}) - E(X) = E(X^{2}) - \mu$$

and since $$V(X) = E[(X- \mu)^{2}] = E(X^{2}) - \mu^2$$

$$V(X) = E(X(X-1)) + \mu -\mu^{2}$$