for a given data M + x bar =32,and M-xbar =2, find z
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Answer:
The random variables X1, X2, ..., Xn are called a random sample of size n
from the population f(x) if X1, X2, ..., Xn are mutually independent random variables and the mar-
ginal probability density function of each Xi is the same function of f(x). Alternatively,
X1, X2, ..., Xn are called independent and identically distributed random variables with pdf f(x).
We abbreviate independent and identically distributed as iid.
Most experiments involve n >1 repeated observations on a particular variable, the first observa-
tion is X1, the second is X2, and so on. Each Xi is an observation on the same variable and each Xi
has a marginal distribution given by f(x). Given that the observations are collected in such a way
that the value of one observation has no effect or relationship with any of the other observations,
the X1, X2, ..., Xn are mutually independent. Therefore we can write the joint probability density
for the sample X1, X2, ..., Xn as
f(x1, x2, ..., xn) = f(x1)f(x2) ··· f(xn) = Yn
i=1
f(xi) (1)
If the underlying probability model is parameterized by θ, then we can also write
f(x1, x2, ..., xn|θ) = Yn
i=1
f(xi|θ) (2)
Note that the same θ is used in each term of the product, or in each marginal density. A different
value of θ would lead to a different properties for the random sample.
1.2. Statistics. Let X1, X2, ..., Xn be a random sample of size n from a population and let
T (x1, x2, ..., xn) be a real valued or vector valued function whose domain includes the sample space
of (X1, X2, ..., Xn). Then the random variable or random vector Y = (X1, X2, ..., Xn) is called a
statistic. A statistic is a map from the sample space of (X1, X2, ..., Xn) call it X, to some space of
values, usually R1 or Rn. T is what we compute when we observe the random variable X take on
some specific values in a sample. The probability distribution of a statistic Y = T(X) is called the
sampling distribution of Y. Notice that T(·) is a function of sample values only, it does not depend
on any underlying parameters, θ.