1. Parameter, Statistic and Random Samples
A parameter is a number that describes the population. It is a fixed
number, but in practice we do not know its value.
A statistic is a function of the sample data, i.e., it is a quantity
whose value can be calculated from the sample data. It is a random
variable with a distribution function. Statistics are used to make inference about unknown population parameters.
The random variables X1, X2,…, Xn are said to form a (simple)
random sample of size n if the Xi’s are independent random
variables and each Xi has the sample probability distribution. We say
that the Xi’s are iid.
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2. Example – Sample Mean and Variance
Suppose X1, X2,…, Xn is a random sample of size n from a population with mean μ and variance σ2.
The sample mean is defined as
The sample variance is defined as
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3. Goals of Statistics Estimate unknown parameters μ and σ2.
Measure errors of these estimates.
Test whether sample gives evidence that parameters are (or are not) equal to a certain value.
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4. Sampling Distribution of a Statistic
The sampling distribution of a statistic is the distribution of values
taken by the statistic in all possible samples of the same size from
the same population.
The distribution function of a statistic is NOT the same as the
distribution of the original population that generated the original
sample.
The form of the theoretical sampling distribution of a statistic will depend upon the distribution of the observable random variables in the sample.
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5. Sampling from Normal population
Often we assume the random sample X1, X2,…Xn is from a normal population with unknown mean μ and variance σ2.
Suppose we are interested in estimating μ and testing whether it is equal to a certain value. For this we need to know the probability distribution of the estimator of μ.
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6. Claim
Suppose X1, X2,…Xn are i.i.d normal random variables with unknown mean μ and variance σ2 then
Proof:
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7. Recall - The Chi Square distribution
If Z ~ N(0,1) then, X = Z2 has a Chi-Square distribution with parameter 1, i.e.,
Can proof this using change of variable theorem for univariate random variables.
The moment generating function of X is
If , all independent then
Proof…
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8. Claim Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ
and variance σ2. Then, are independent standard normal
variables, where i = 1, 2, …, n and
Proof: …
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9. t distribution Suppose Z ~ N(0,1) independent of X ~ χ2(n). Then,
Proof:
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10. Claim
Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ and variance σ2. Then,
Proof:
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11. F distribution
Suppose X ~ χ2(n) independent of Y ~ χ2(m). Then,
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12. Properties of the F distribution
The F-distribution is a right skewed distribution.
i.e.
Can use Table 7 on page 796 to find percentile of the F- distribution.
Example…
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13. Recall - The Central Limit Theorem
Let X1, X2,…be a sequence of i.i.d random variables with mean
E(Xi) = μ < ∞ and Var(Xi) = σ2 < ∞. Let
Then, converges in distribution to Z ~ N(0,1).
Also, converges in distribution to Z ~ N(0,1).
Example…
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