The M1M1 treasure hunt, week 4
Chapter 2: Talkin' 'bout my generation
Ann knocked once more on the door of the strange blue box.
"Come in. Well? What's the maximum value of a?" demanded her tutor.
"Er, 3, I think," she essayed uncertainly.
"Wrong. Thank goodness. I thought you might have decided to work really hard to get it right even though I told you last week you'd got it wrong, and so upset the fragile nature of Causality. We time travellers have to be very careful."
"I did try!" Ann protested. "But I'm worried that if I travel into the future and find all the answers, won't that count as plagiarism?"
"Hmm. I suppose that could be a problem. Maybe we should stick with the past for now. When would you like to go?"
"I'm ready now," she answered eagerly.
"NO, NO, NO. When IN THE PAST would you like to visit?" snapped the irrascible Dr Hu.
"Oh, I see. Well, I'm quite keen on M1S1. Could we travel to the birth of Statistics?"
The doctor made a few calculations, pressed some buttons, pulled some levers and turned on the light switch. Tentatively Ann opened the door.
"It's a girl, Mrs McCoy!" exclaimed a voice outside. And indeed, in front of her, Ann could see a baby girl, precociously sucking her thumb. As Ann watched, the baby removed her thumb and stared at her hands, as if trying to calculate in how many different ways she could insert N fingers from one hand and M from the other into her tiny mouth, with or without replacement.
"I didn't mean a real birth - I meant the beginning of Stats as a subject!" hissed Ann, more than a little embarrassed.
"Oh," mused the doctor, quietly shutting the door to the hospital room. "You mean like where does Statistics come from? What generates probability? Well, you need a probability generating function, of course. Functions define everything. Choose a positive integer at random."
"I can't do that!" cried Ann. "There's an infinity of them."
"Yes you can - they don't all have to have an equal chance.
Suppose you choose the integer, n, with probability pn. Then we can define the function:
f(x) = ∑pnxn
That tells you everything you need to know. Suppose f(x) = 4/(10 - 7x + x2). Then what's the probability p2 you'll choose 2, (written e.g. as 0.234)?"
"So does every function correspond to a probability distribution?" asked Ann.
"My goodness, no. First of all it has to have a power series expansion. Then all the coefficients must be non-negative. And they all have to sum to one, which means f(1) = 1. This means the series has to converge at x = 1, and so its radius of convergence must be at least 1. In fact, the series for the f(x) we just wrote down has the radius of convergence
"But," began Ann slowly, "if I take two series with positive coefficients and multply them together, I'll get another series with positive coefficients. And the business at x = 1 works too. So if I have two of these generating functions and multiply them together, I'll get a third."
"Very good. And if you think about it, you'll see that this gives you the distribution of the sum of the outcomes of trials using the first two functions."
"That works? How cool. So what if we take a function and multiply it by itself very many times. What distribution do we approach then?"
The doctor felt a bit out of his depth, though he suspected the question had a normal answer. "Let's leave that for her when she's older," he replied, indicating the closed door. "Time we were getting back. But you could consider another generating function, g(x) = ex-1 and tell me the chance of getting a sum of 2 using f(x) and g(x) once each."
It felt a bit fishy, but it was a simple matter to find the series for the product fg.