# Probability density function and distribution from samples to population

Heya got a small doubt here

most of the discrete random variables utilize simple examples of situations like 3 coin toss or 2 dice rolls or girl/boys born. These examples are simple in the sense that we can chart out the entire sample space and discrete values of the random variable. We can then have a complete frequency or probability distribution.

but what happens in situations when it is not possible to prepare the complete sample space. say I note down childbirths in a hospital for 30 days. Can I use this sample and convert it to proportions and give out the probability of no of childbirths on a given date? (stupid example I know)

Hey hopefully I'm understanding your question correctly and this helps. It sounds like this is a general theory question about how to model a real-world event when we don't have access to the full population. This is the case most of the time, which is why we rely on probability distributions.

In your example, you want to predict either the number of childbirths on a given date or the probability of a certain number of childbirths on that date. The simplest way (that I can think of) to model this would be to assume that the number of childbirths a day follows a Binomial distribution:

i.e. Let X be the random variable "the number of childbirths a day", following a Binomial distribution with unknown n and p parameters: X ~ Binomial(n, p)

The question is how to estimate the values n (number of women who could give birth a day) and p (probability of giving birth that day). This is known as statistical inference and there are many available frequentist and Bayesian methods for performing this.