X ~ N(u, ) 3) Central Limit theorem, is the most important theorem in statistics. Proven by many mathematicians and computer scientist in different settings, form, Lindeberg to Alan Turing (The founding father of the Artificial Intelligence), and having many application in almost all fields it gained importance in the world of research. Assume X1,.,Xn are i.i.d (independently and identically distributed) random variables from an unknown distribution with mean and variance & o. Items of this questions start with a bold font. a)Show that for the sample mean which is a random variable itself E(X) = 4 & V(X) (b)Without loss of generality assume Y1,...Yn, are standardized version of X;. Which means mean and variance of them are 0 and 1. You can Check to see mean and variance of Y = X- are 0 and 1. (in previous homework we checked it) Also it is an easy algebra to check that - b)question: Using 2.c) show that M (t)= My-o (t) = (My (- (c)take the limit of right hand side of b) and using 2. b and fact that Y; have mean 0 and variance 1 i.i.d, (with L'hopital) show that D NO DISTRBUT X - u lim M, (t) lim My-(t) = lim (My,() n00 d)Since this limit matches 2.d) X - E(X) Var(X) VIn Y -0 4, N(0,1) %3D This is basically the statement of the Central Limit Theorem : N(0, 1) X ~ N(µ,-) or Multiplying by n, we also have To - E(T) To - nu Vno VVar(To) N(0, 1) or where DISTRIBE