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Some Theory Problems (Part II) Problem 1 Let X1 ~ N(p.a²) and X2 w N(A,O2), where X1 and X2 are independent. Compute 91-0<1-1,<0). Problem 2 2.i For bivariate random variable (X,Y), derive the shortcut formula for covariance, i.e., show = 2.ii If random variables X and Y are independent, what does E(XY] equal? Problem 3 Definition: Continious random variables X and Y are independent if their joint densify f(x.g) oun be expressed as the product of their marginal densifies x(x) and fr(g), i.e., 3.i If X and Y are independent continuous random variables, show that E(XY] = E|X): E(Y]. (Prove this result using the definition of expectation for continuous random variables and not by referencing problem 2.ii.) 3.ii Now assume (X,Y) is distributed bivarinte normal. If the linent correlation coeficient P = 0, show X and Y are independent normal randorn variables. Problem 4 Let X1 w Binomial(m.p) and X2 ~ Binomial(n.p) and consider the estimator X1+X2 P m Prove that P is an unbiased estimator of p. Lab 4 covers sampling distributions, the central limit theorem (CIT), and maximum likelihood estimation (MLE). For ench exercise we assume X1,X2, X. is a randorn sample from a Poissom distribution with rate parameter & 0. Section I: CLT and Sum of Independent Poisson Random Variables Set-up Part I: Equivalent Forms of the CLT Assuming X1,X2 X. is a random sample from some distribution with common mean e and variance of. then for sufficiently large n: CLT; N(H,O) CITHi X=-P N(0,1) CLTHi Tn CITi N(0,1) Expressions (CLT ii) and (CLT. iv) are oftem more desirable because there is mo sample size (n) om the right hand side of the relation. Recall that the CLT is an asymptotic result (n - 00), hence the sample size should not appear after taking the limit. 1 Set-up Part II: Sum of Independent Poisson Random Variables With some work, one can show that if X1 X2 X2 = Poisson( A). then the sum of X: is also a Poisson randorn variable with rate nd. i.e., i=1 1 Application Suppose that the number of customers arriving at a gas station (X;) in any given day has a Poisson distribution with rate A = 100 [customers per day|- The manager of the gas statiom is interested in the total number of customers (T. = Li-1 X1) arriving in one week and one month Assume that one weck has n = 7 days and ome month has n = 31 days. Solve the following problems: (1)-(2) 1. Compute the probability (and approximate probability) that more than 670 customers visit the gas station in a weck. Solve this problem using the Poisson distribution and the CLT. Compare the approximate answer to the correct Poissom probability. (Note that we are assuming n = 7 < 30 for the CLT, which is typically discouraged) 2. Compute the probability (and approximate probability) that more than 3000 customers visit the gas station in a month. Solve this problem using the Poisson distribution and the CLT. Compare the approximate answer to the correct Poisson probability. Note the following: 670 e-200(700)* 3000 0.0364 le! e *=0 Section II: MLE of Poisson Rate Parameter Let X1, X2, X. be a random sample from a Poisson distribution with rate parameter A> 0 Solve the following problems: (3)-(6) 3. Derive the maximum likelihood estimator of A. Denote the MLE by is 4. Show that & is an unbiased estimator of & 5. Derive an expression for the standard error of is 6. Identify the large sample distribution of & 2

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