6.1 Random Samples
Idea: have some population with unknown distribution of some
quantity ? say, heights of students in class
Note: If population is finite, and you could measure values
for the whole population, could compute these parameters directly.
Suppose have m individuals in population, and their heights
are x1, x2, ..., xm.
the (unknown) mean, variance, and moments of the quantity are called the
finite population mean:
finite population variance:
average squared deviation from mean!
Approach: choose a sample of n objects from population; use
the values for the items in the sample to estimate the values of the parameters
for whole population
usually, hard to get values for whole population (too hard to collect all
if population is infinite, e.g., distribution is continuous, clearly canít
measure whole population!
Thus get the following definition:
pick at random n objects from population
value of interest for each depends on which object selected, i.e., is a
random variable whose distribution is that of whole populaton
thus have n random variables X1, X2,
..., Xn with identical (but unknown) distribution
assume the values of the random variables are independent, i.e., that the
value of one doesn't affect the value of another
Def: A random sample of size n from a particular
distribution is a set of n independent random variables X1,
X2, ..., Xn , each of which has this same distribution.
when we choose a particular sample, get an observed value for each of the
denote observed values for the sample by x1, x2,
..., xn (small x's)
use these values to estimate parameters for population
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