3
$\begingroup$

Imagine that there are $N$ balls in a box, among which $m$ are white and $N-m$ are black. If one blindly picks a ball in the box, the probability that the ball is white is simply:

$$p_w = \frac{m}{N}$$

Now, Imagine that $m$ is not a constant, but the outcome of some random process; as a consequence, $p_w$ becomes a random variable, whose distribution can be derived from the distribution of $m$.

The very concept of distribution of a probability seems strange to me. In a general way, is it fine to have a probability as a random variable? Can one get a grasp on what that means and what it implies?

In this precise situation, can one still define the probability to pick a white ball, or is the best one can do is to use the expected value of getting a white ball?

$\endgroup$
  • 1
    $\begingroup$ Probabilities as random variables come up all the time in a wide variety of contexts. For example, imagine you you're trying to model probability of making a claim on a third-party property damage policy in car insurance (insuring against stuff like crashing into someone else's possessions house, car, etc). Individuals will each have different probability of generating a claim (affected by how good a driver they are, how much they drive, the condition of car they have, etc), so the probability of a claim a one-year insurance policy is itself a random variable from some distribution $\endgroup$ – Glen_b -Reinstate Monica Aug 26 '17 at 1:00
6
$\begingroup$

A probability that's randomly generated is a perfectly legitimate thing to have in a model. Such a model is an example of a hierarchical model: a model in which some parameters are themselves treated as random variables. Hierarchical models come up most frequently in Bayesian statistics, but they see a lot of use in frequentist statistics, too, especially in the form of so-called random effects, which are regression coefficients that are modeled as random variables.

In your example, the marginal probability of picking a white ball at any given time will be the mean of $p_w$. In Bayesian terms, if the specified distribution of $p_w$ is its prior, then the marginal probability of picking a white ball will be the posterior mean of $p_w$.

$\endgroup$
2
$\begingroup$

Probabilities as random variables come up all the time in a wide variety of contexts.

For one example, imagine you you're trying to model probability of making a claim on a third-party property damage policy in car insurance (insuring against stuff like crashing into someone else's possessions house, car, etc). Individuals will each have different probability of generating a claim (affected by how good a driver they are, how much they drive, the condition of car they have, etc), so if you take a random policy it corresponds to taking a random member of the population of insured's -- the probability of a claim a one-year insurance policy is itself a random variable from some distribution.

(I'm ignoring things like people driving other people's cars here. Still, it conveys the idea.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.