week11 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 X 1, X 2,…, X n are said to form a (simple) random sample of size n if the X i ’s are independent random variables and each X i has the sample probability distribution. We say that the X i ’s are iid.

week12 Example – Sample Mean and Variance Suppose X 1, X 2,…, X n is a random sample of size n from a population with mean μ and variance σ 2. The sample mean is defined as The sample variance is defined as

week13 Goals of Statistics Estimate unknown parameters μ and σ 2. Measure errors of these estimates. Test whether sample gives evidence that parameters are (or are not) equal to a certain value.

week14 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.

week15 Sampling from Normal population Often we assume the random sample X 1, X 2,…X n is from a normal population with unknown mean μ and variance σ 2. 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.

week16 Claim Suppose X 1, X 2,…X n are i.i.d normal random variables with unknown mean μ and variance σ 2 then Proof:

week17 Recall - The Chi Square distribution If Z ~ N(0,1) then, X = Z 2 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…

week18 Claim Suppose X 1, X 2,…X n are i.i.d normal random variables with mean μ and variance σ 2. Then, are independent standard normal variables, where i = 1, 2, …, n and Proof: …

week19 t distribution Suppose Z ~ N(0,1) independent of X ~ χ 2 (n). Then, Proof:

week110 Claim Suppose X 1, X 2,…X n are i.i.d normal random variables with mean μ and variance σ 2. Then, Proof:

week111 F distribution Suppose X ~ χ 2 (n) independent of Y ~ χ 2 (m). Then,

week112 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…

week113 The Central Limit Theorem Let X 1, X 2,…be a sequence of i.i.d random variables with E(X i ) = μ < ∞ and Var(X i ) = σ 2 < ∞. Let Then, for - ∞ < x < ∞ where Z is a standard normal random variable and Ф(z)is the cdf for the standard normal distribution. This is equivalent to saying that converges in distribution to Z ~ N(0,1). Also, i.e. converges in distribution to Z ~ N(0,1).

week114 Example Suppose X 1, X 2,…are i.i.d random variables and each has the Poisson(3) distribution. So E(X i ) = V(X i ) = 3. The CLT says that as n  ∞.

week115 Examples A very common application of the CLT is the Normal approximation to the Binomial distribution. Suppose X 1, X 2,…are i.i.d random variables and each has the Bernoulli(p) distribution. So E(X i ) = p and V(X i ) = p(1- p). The CLT says that as n  ∞. Let Y n = X 1 + … + X n then Y n has a Binomial(n, p) distribution. So for large n, Suppose we flip a biased coin 1000 times and the probability of heads on any one toss is 0.6. Find the probability of getting at least 550 heads. Suppose we toss a coin 100 times and observed 60 heads. Is the coin fair?