The big advantage of sampling with replacement (the above procedure) is that $X_i$'s will be independent and this makes the analysis much simpler. However, if the population is large, then the probability of choosing one person twice is extremely low, and it can be shown that the results obtained from sampling with replacement are very close to the results obtained using sampling without replacement. You might ask why do we do the sampling with replacement? In practice, we often do the sampling without replacement, that is, we do not allow one person to be chosen twice. In general, $X_i$ is the height of the $i$th person that is chosen uniformly and independently from the population.Again, every person in the population has the same chance of being chosen. To determine the value of $X_2$, again we choose a person uniformly (and independently from the first person) at random and let $X_2$ be the height of that person. A sample where every member has an equal probability of being chosen is called a simple random sample.Here, every person in the population has the same chance of being chosen. We chose a person uniformly at random from the population and let $X_1$ be the height of that person.To do this, we define random variables $X_1$, $X_2$, $X_3$, $.$, $X_n$ as follows: We choose a random sample of size $n$ with replacement from the population and let $X_i$ be the height of the $i$th chosen person. For example, suppose that our goal is to investigate the height distribution of people in a well defined population (i.e., adults between 25 and 50 in a certain country). A person, object, or some other well-defined item upon which a treatment is applied. The sample is then called a simple random sample. There are other types of sampling, such as stratified and cluster sampling. In SRS, every element has the same probability sampling. A simple random sample (SRS) is a statistical sampling technique that uses chance as a factor to decide which data points are included in the sample. When collecting data, we often make several observations on a random variable. A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring. A sample is a subset of data selected from an entire population.
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