I am a newbie to statistics and found this.
In statistics, θ, the lowercase Greek letter ‘theta’, is the usual
name for a (vector of) parameter(s) of some general probability
distribution. A common problem is to find the value(s) of theta.
Notice that there isn’t any meaning in naming a parameter this way. We
might as well call it anything else. In fact, a lot of distributions
have parameters which are usually given other names. For example, it
is common use to name the mean and deviation of the normal
distribution μ (read: ‘mu’) and deviation σ (‘sigma’), respectively.
But I still don’t know what that means in plain English?
It is not a convention, but quite often θ stands for the set of parameters of a distribution.
That was it for plain English, let’s show examples instead.
Example 1. You want to study the throw of an old fashioned thumbtack (the ones with a big circular bottom). You assume that the probability that it falls point down is an unknown value that you call θ. You could call a random variable X and say that X=1 when the thumbtack falls point down and X=0 when it falls point up. You would write the model
and you would be interested in estimating θ (here, the proability that the thumbtack falls point down).
Example 2. You want to study the disintegration of a radioactive atom. Based on the literature, you know that the amount of radioactivity decreases exponentially, so you decide to model the time to disintegration with an exponential distribution. If t is the time to disintegration, the model is
Here f(t) is a probability density, which means that the probability that the atom disintegrates in the time interval (t,t+dt) is f(t)dt. Again, you will be interested in estimating θ (here, the disintegration rate).
Example 3. You want to study the precision of a weighing instrument. Based on the literature, you know that the measurement are Gaussian so you decide to model the weighing of a standard 1 kg object as
Here x is the measure given by the scale, f(x) is the density of probability, and the parameters are μ and σ, so θ=(μ,σ). The paramter μ is the target weight (the scale is biased if μ≠1), and σ is the standard deviation of the measure every time you weigh the object. Again, you will be interested in estimating θ (here, the bias and the imprecision of the scale).