Question

Let T1,T2,…,Tn be i.i.d. observations, each drawn from a common normal distribution with mean zero. With probability 1/2 this normal distribution has variance 1, and with probability 1/2 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

∣c1*∑(i=1 to n)ti^2 + 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)'.