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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. • 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 1 Example • Toss a coin n times. • Suppose 1 Xi 0 if i th toss came up H if i th toss came up T • Xi’s are Bernoulli random variables with p = ½ and E(Xi) = ½. 1 n • The proportion of heads is X n X i . It is a statistic. n i 1 week1 2 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. • Probability rules can be used to obtain the distribution of a statistic provided that it is a “simple” function of the Xi’s and either there are relatively few different values in he population or else the population distribution has a “nice” form. • Alternatively, we can perform a simulation experiment to obtain information about the sampling distribution of a statistic. week1 3 Markov’s Inequality • If X is a non-negative random variable with E(X) < ∞ and a >0 then, P X a EX a Proof: week1 4 Chebyshev’s Inequality • For a random variable X with E(X) < ∞ and V(X) < ∞, for any a >0 P X E X a V X a2 • Proof: week1 5 Law of Large Numbers • Interested in sequence of random variables X1, X2, X3,… such that the random variables are independent and identically distributed (i.i.d). Let 1 n Xn Xi n i 1 Suppose E(Xi) = μ , V(Xi) = σ2, then 1 n 1 n E X n E X i E X i n i 1 n i 1 and 1 n 1 V X n V X i 2 n i 1 n n V X i 1 i 2 n • Intuitively, as n ∞, V X n 0 so X n E X n week1 6 • Formally, the Weak Law of Large Numbers (WLLN) states the following: • Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , V(Xi) = σ2 < ∞, then for any positive number a P Xn a 0 as n ∞ . This is called Convergence in Probability. Proof: week1 7 Example • Flip a coin 10,000 times. Let 1 Xi 0 if i th toss came up H if i th toss came up T • E(Xi) = ½ and V(Xi) = ¼ . • Take a = 0.01, then by Chebyshev’s Inequality 1 1 1 1 P X n 0.01 2 2 4 410,000 0.01 • Chebyshev Inequality gives a very weak upper bound. • Chebyshev Inequality works regardless of the distribution of the Xi’s. • The WLLN state that the proportions of heads in the 10,000 tosses converge in probability to 0.5. week1 8 Strong Law of Large Number • Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , then X n converges to μ as n ∞ with probability 1. That is 1 P lim X 1 X 2 X n 1 n n • This is called convergence almost surely. week1 9 Central Limit Theorem • The central limit theorem is concerned with the limiting property of sums of random variables. • If X1, X2,…is a sequencen of i.i.d random variables with mean μ and variance σ2 and , S X n i 1 i then by the WLLN we have that Sn in probability. n • The CLT concerned not just with the fact of convergence but how Sn /n fluctuates around μ. • Note that E(Sn) = nμ and V(Sn) = nσ2. The standardized version of Sn is S n n Zn and we have that E(Zn) = 0, V(Zn) = 1. n week1 10 The Central Limit Theorem • Let X1, X2,…be a sequence of i.i.d random variables with E(Xi) = μ < ∞ n and Var(Xi) = σ2 < ∞. Let S n X i i 1 S n n lim P z PZ z z for - ∞ < x < ∞ Then, n n 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 Z n Z ~ N(0,1). • S n n converges in distribution to n Xn x lim P x Also, n n i.e. Z n Xn converges in distribution to Z ~ N(0,1). n week1 11 Example • Suppose X1, X2,…are i.i.d random variables and each has the Poisson(3) distribution. So E(Xi) = V(Xi) = 3. • The CLT says that P X 1 X n 3n x 3n x as n ∞. week1 12 Examples • A very common application of the CLT is the Normal approximation to the Binomial distribution. • Suppose X1, X2,…are i.i.d random variables and each has the Bernoulli(p) distribution. So E(Xi) = p and V(Xi) = p(1- p). • The CLT says that P X 1 X n np x np1 p x as n ∞. • Let Yn = X1 + … + Xn then Yn has a Binomial(n, p) distribution. So for large n, PYn y P Yn np np1 p y np y np np1 p np1 p • 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? week1 13 Sampling from Normal Population • If the original population has a normal distribution, the sample mean is also normally distributed. We don’t need the CLT in this case. • In general, if X1, X2,…, Xn i.i.d N(μ, σ2) then Sn = X1+ X2+…+ Xn ~ N(nμ, nσ2) and 2 Sn Xn ~ N , n n week1 14 Example week1 15