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PARAMETER AND STATISTIC
A population, as described in Section 2.4 in XI Standard text book, is a collection of units/objects/numbers under study, whose elements can be considered as the values of a random variable, say, X. As mentioned in Section 9.3 in XI Standard text book, there will be a probability distribution associated with X.
Parameter: Generally, parameter is a quantitative characteristic, which indexes/identifies the respective distribution. In many cases, statistical quantitative characteristics calculated based on all the units in the population are the respective parameters. For example, population mean, population standard deviation, population proportion are parameters for some distributions.
Recall: The unknown constants which appear in the probability density function or probability mass function of the random variable X, are also called parameters of the corresponding distribution/population.
The parameters are commonly denoted by Greek letters. In Statistical Inference, some or all the parameters of a population are assumed to be unknown.
Random sample: Any set of reliazations (X1, X2 , ..., Xn) made on X under independent and identical conditions is called a random sample.
Statistic: Any statistical quantity calculated on the basis of the random sample is called a statistic. The sample mean, sample standard deviation, sample proportion etc., are called statistics (plural form of statistic).They will be denoted by Roman letters.
Let (x1, x2, …, xn) be an observed value of (X1, X2, ..., Xn). The collection of (x1, x2, …, xn) is known as sample space, which will be denoted by ‘S’.
A set of n sample observations can be made on X, say, x1, x2, …, xn for making inferences on the unknown parameters. It is to be noted that these n values may vary from sample to sample. Thus, these values can be considered as the realizations of the random variables X1, X2, ..., Xn ,which are assumed to be independent and have the same distribution as that of X. These are also called independently and identically distributed (iid) random variables.
In Statistical Inference, the sample standard deviation is defined as S = where . It may be noted that the divisor is n – 1 instead of n
The statistic itself is a random variable, until the numerical values of X1, X2, ..., Xn are observed, and hence it has a probability distribution.
Notations to denote various population parameters and their corresponding sample statistics are listed in Table 1.1. The notations will be used in the first four chapters of this book with the same meaning for the sake of uniformity.
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