The theory of probability pertains to the various possible outcomes that might be obtained and the possible events that might occur when an experiment is performed.
Probability is defined over the concept of event and experiment, which relates to the definition of probability, not an estimation, population not sample.
Experiment and Event. An experiment is any process, real or hypothetical, in which the possible outcomes can be identifified ahead of time. An event is a well-defifined set of possible outcomes of the experiment.
Sample Space. The collection of all possible outcomes of an experiment is called the sample space of the experiment.
The definition of sample space forms under the idea that the experiment is undertaken.
it is viewed that all the possible outcomes of the event form the sample space, and each outcome is a sample of the event. All sample forms the population of the event. It means different from the 'sample' of a population.
when the experiment is selected from a sample, it can be viewed that the experiment is chosen from a sample from the population. thus the outcomes happen to be the sample of the population, and it's called the sample space.
Definition 3.1.1 Random Variable. Let S be the sample space for an experiment. A real-valued function that is defifined on S is called a random variable.
The random variable is actually a function on the sample space, which defined on an event or experiment.
Random Samples/i.i.d./Sample Size. Consider a given probability distribution on the real line that can be represented by either a p.f. or a p.d.f. f . It is said that n random variables X1,...,Xn form a random sample from this distribution if these random variables are independent and the marginal p.f. or p.d.f. of each of them is f . Such random variables are also said to be independent and identically distributed, abbreviated i.i.d.We refer to the number n of random variables as the sample size.
Random variable are defined on the sample space as the probability.
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