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. week1

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 week1

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. week1

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. week1

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. week1

Claim Suppose X1, X2,…Xn are i.i.d normal random variables with unknown mean μ and variance σ2 then Proof: week1

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… week1

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: … week1

t distribution Suppose Z ~ N(0,1) independent of X ~ χ2(n). Then, Proof: week1

Claim Suppose X1, X2,…Xn are i.i.d normal random variables with mean μ and variance σ2. Then, Proof: week1

F distribution Suppose X ~ χ2(n) independent of Y ~ χ2(m). Then, week1

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… week1

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… week1