6. Find the most popular health center(s) (in orders up-to-date.

Problem 3 (40 points) Let {X, } be any sequence of random variables. Let Yr = _ EF, X,. A necessary and sufficient condition for the sequence {X, } to satisfy the weak law of large numbers is that E n -+ 0 as n -+ 0o 1 + Y2 First show that for any two positive numbers a, b, a 2 b > 0 a 1+b 1 ta b 21Problem 4 Weak Law of Large numbers. (a) [easy] Given X1. .... X,, ~ with finite a and o', show that X,, 4 u, the "weak" weak law of large numbers.Problem 5. Let X1, X2, . .. be an infinite sequence of Lid. random variables. Let f be a function. Prove the weak law of large numbers for {f(X,.)}. Specifically, show that if Sa = f(XI ) + f (X2 ) + ... + / (X.). then for any 0. P Sn You need to prove this result from scratch. You cannot use the law of large numbers for the sequence {X.). You canProblem 8 (Weak Law of Large Numbers). In this problem, we prove the so called weak law of large numbers following a series of steps. Let {X,} be a sequence of discrete independent and identically distributed random variables with E[X1] = p and variance Var( X1) = 02. Identically distributed discrete random variables have the same pmf. Thus, all the random variables have the same expected value and variance. Define Sn = X1+ X2 + ... + Xn n The weak law of large number (WLLN) states that for any 0, P Sn - A > E -+0, as n -+ 00. To prove this, complete the following steps. 1. Show that E[Sn] = nu. This means that E [Sa ] = /. 2. Next show that Var On n n 3. Use Chebyshev's inequality to show that P ( " - 1 / ze ) 5 Var( Swin) 4. Now, substitute the expression for Var (So ) to prove the result