Is the chance of a variable also a parameter for a probability distribution?

Question

I'm new to statistics and I'm a bit confused about the concepts of 'chance of a variable' and 'parameters of a probability distribition'. Is chance also a parameter? And if so: can computing the chance of a variable be considered as an estimation for the parameters?

Answer 1

Maybe you are new enough to statistics that you've only thought about normal and binomial distributions.

BINOMIAL. A binomial random variable $X$ counts the number of Successes in a simple experiment. Perhaps $X$ is the number of times you get a 6 when you roll a die twice. There are two parameters of the distribution of this random variable. The first is $n = 2$, the number of independent 'trials' (here rolls of the die). The second is the probability $p = 1/6,$ which is the probability (or 'chance') that you get a six on any one roll. But it is not ALWAYS the case that a parameter is the 'chance' of anything.

From this information we can derive the distribution of $X$, which gives probabilities $P(X = 0) = (5/6)^2 = 25/36,\; P(X = 1) = 2(1/6)(5/6) = 10/36$ and $P(X = 2) = (1/6)^2 = 1/36.$ Notice that summing the probabilities of each of the possible values of $X$ must always give 1. Here, we have $10/36 + 25/36 + 1/36 = 1.$ One can prove that the mean $\mu$ of this distribution is $1/3$ and that the standard deviation is $\sigma = 0.6202.$

NORMAL. A normal random variable $Y$ also has two parameters, the mean $\mu$ (or center) of the distribution, and the standard deviation $\sigma$, which says how spread out the values are likely to be.

This is a continuous random variable, so it is not possible to list the values it will take. Instead of a list of individual probabilities, we give a density function. The total area under the density function is is 1. For example, if $\mu = 100$ and $\sigma = 15$ then it turns out that $P(Y > 100) = 1/2 = 50\%$; half of the area under the density curve is to the right of 1/2. Also, $P(Y > 130) = 0.02275$ (just under $2.5\%$). (Perhaps this random variable models IQ scores; there are not many people with IQs above 130.)

In this case the mean and standard deviation are the same as the parameters, but neither of them has any direct interpretation as the 'chance' of anything.

MORE GENERAL. As you see additional random variables and their distributions, it is natural to try to see if the parameters have any intuitive interpretation. Often the answer is YES for the most commonly-used distributions, and it is worthwhile to try to understand the intuitive connection. It may have to do with a 'chance', a 'rate', something to do with the shape of the distribution, or something else. There is no general rule for this. (Also, sometimes the parameters are very abstract and any kind of intuitive interpretation is difficult.)

Below are plots of the two distributions mentioned as examples above. A vertical red line shows the location of the mean in each case.