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FAQ 1.1: What is a random process?
Q: So a parameter is a numerical property of a process… but what is the process? All I have is Buzz’s 16
attempts.
A: We can think of Buzz’s 16 attempts as a sample from his long-run selection process. He has some
underlying probability of pushing the correct button, but he’s unlikely to be correct every time, even if the
dolphins are communicating. There is randomness in his choices. His choices might be affected by Doris’
squeaks, but also by how hungry he is, how tired he is, how warm the water is. We are trying to see how
much Doris’ communication is influencing him. Everything else gets lumped together into “random
chance.” Any one outcome from a random process is unknown to us in advance (like tossing a coin), but
if we observe outcomes from the process for long enough, we should start to see some patterns amidst
the randomness. So you can think of a random process as an unlimited source of potential observations
for your sample.
Q: Can I use any old set of observations from a process as a sample?
A: Some samples are good and others are terrible.
Q: How do I tell the good apples from the rotten ones?
A: For a Yes/No process like Buzz’s selections, we need to be willing to make certain assumptions along
the way. For example, we assumed Buzz’s probability of a correct guess was the same every time. In
fact, under the null hypothesis, we assumed it was 50/50 every time. But even if the probability is not 0.5,
we are assuming it is not changing. In other words, we assume that Buzz doesn’t get better through
learning and that he doesn’t get tired or bored or quit trying. We are also assuming that his previous trials
don’t influence his future guesses – like a coin, he would have no memory of his past guesses.
Q: So how do I know if I have a good sample?
A: You need to believe that nothing about the process is changing and that each outcome does not
depend on past outcomes. If you’re a coin, your chance of heads doesn’t change, and you have no
memory of how you’ve landed in the past. If this is true about the process, then you will have a good
sample! On the other hand, if Buzz’s probability of success is different in the morning than in the
afternoon, then only observing him in the morning will not give you a good representation of his overall
probability of success.
Q: So then how does the parameter come into play?
A: Parameter is a hard concept, precisely because we never see it! For the dolphin example, we want to
know whether Buzz is just guessing. We think of his guesses as a potentially never-ending process (like
coin tossing), and the parameter is the probability that Buzz will be correct in his next attempt. As you saw
in the Preliminaries, this means if we were to observe the random process forever, this probability is the
long-run proportion of times that Buzz pushes the correct button.
Q: So I see why the parameter isn’t just 15/16, that is his sample proportion and we may not have hit that
long-run probability yet. So the parameter is just 0.5?
A: Not necessarily. The parameter equals 0.5 if he is just guessing. That’s the chance model we
simulated. But his probability may actually be different from 0.5.
Q: So the parameter is his actual probability of pushing the correct button. It’s a number, but we don’t
know its value.
A: Correct! But keep in mind that we will define lots of different kinds of parameters in future chapters.
The “probability of success” parameter only applies in this type of Yes/No random process.
June 27, 2014
MAA PREP workshop
18