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