Conditioned on the result of an unbiased coin flip, the random variables T1,T2,…,Tn are independent and identically distributed, each drawn from a common normal distribution with mean zero. If the result of the coin flip is Heads, this normal distribution has variance 1 ; otherwise, it has variance 4 . Based on the observed values t1,t2,…,tn , we use the MAP rule to decide whether the normal distribution from which they were drawn has variance 1 or variance 4 . The MAP rule decides that the underlying normal distribution has variance 1 if and only if

Question

|c1*∑ (i=1 to n)t^2i+c2*∑ (i=1 to n)ti | < 1.

Find the values of c1≥0 and c2≥0 such that this is true. Express your answer in terms of n , and use "ln" to denote the natural logarithm function, as in "ln(3)".

C1 =?

C2 = ?

Answer 1

Any idea?

Answer 2

C2=0

Answer 3

C1 = 3/(8*n*ln(2))