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Consider the problem of sampling sequentially from a finite number of $N \geq2$ populations, specified by random variables $X^i_k$, $ i = 1,\ldots , N,$ and$k = 1, 2, \ldots$; where $X^i_k$ denotes the outcome from population $i$ the$k^{th}$ time it is sampled. It is assumed that for each fixed $i$,  $\{ X^i_k \}_{k \geq 1}$ is a sequence of i.i.d. normal random variables,with unknown mean $\mu_i$ and unknown variance $\sigma_i^2$.  The objective is to have a policy $\pi$ for deciding from which of the $N$populations to sample form at any time $n=1,2,\ldots$ so as to maximize theexpected sum of outcomes of $n$ samples or equivalently to minimize the regretdue to lack on information of the parameters $\mu_i$ and $\sigma_i^2$. In thispaper, we present a simple inflated sample mean (ISM) index policy that isasymptotically optimal in the sense of Theorem 4 below. This resolves astanding open problem from Burnetas and Katehakis (1996). Additionally, finitehorizon regret bounds are given.

Normal Bandits of Unknown Means and Variances: Asymptotic Optimality, Finite Horizon Regret Bounds, and a Solution to an Open Problem