In statistics, θ, the small letter Greek letter "theta", is the usual name for a (vector of) parameter(s) that some basic probability distribution. A common problem is to uncover the value(s) that theta. Notification that there isn"t any an interpretation in specify name a parameter this way. We might also call that anything else. In fact, a many distributions have actually parameters which room usually offered other names. For example, the is usual use to surname the mean and also deviation of the normal circulation μ (read: "mu") and deviation σ ("sigma"), respectively.

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But ns still don"t understand what that way in plain English?

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It is not a convention, however quite frequently $\theta$ means the set of parameters of a distribution.

That was it for level English, let"s present examples instead.

Example 1. You desire to examine the throw of an old fashioned thumbtack (the ones with a large circular bottom). Girlfriend assume that the probability the it falls suggest down is an unknown worth that you call $\theta$. Girlfriend could contact a arbitrarily variable $X$ and also say the $X=1$ when the thumbtack falls allude down and also $X=0$ once it falls point up. You would certainly write the model

$$P(X = 1) = \theta \\P(X = 0) = 1-\theta,$$

and you would be interested in estimating $\theta$ (here, the proability that the thumbtack falls point down).

Example 2. You want to study the disintegration of a radiation atom. Based on the literature, you understand that the lot of radioactivity decreases exponentially, therefore you decision to version the time to fragmentation with one exponential distribution. If $t$ is the moment to disintegration, the model is

$$f(t) = \theta e^-\theta t.$$

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 certainly be interested in estimating $\theta$ (here, the fragmentation rate).

Example 3.

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You want to examine the precision that a weighing instrument. Based upon the literature, you understand that the measurement space Gaussian so you decide to design the weighing the a standard 1 kg thing as

$$f(x) = \frac1\sigma \sqrt2\pi \exp \left\ -\left( \fracx-\mu2\sigma \right)^2\right\.$$

Here $x$ is the measure offered by the scale, $f(x)$ is the thickness of probability, and the parameters are $\mu$ and $\sigma$, for this reason $\theta = (\mu, \sigma)$. The paramter $\mu$ is the target load (the scale is biased if $\mu \neq 1$), and $\sigma$ is the traditional deviation of the measure every time you sweet the object. Again, you will certainly be interested in estimating $\theta$ (here, the bias and the imprecision of the scale).