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)
 A: 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.
The long answer:
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.
The short answer:
We cannot generalise a small sample to a population easily but we can make an assumption about which distribution the model may follow and then use statistical inference to estimate the shape of the distribution.
