The expected value of the sum of two random variables is the sum of their individual expected values. The variance of the sum is the sum of the individual variances plus twice the covariance.
So, if you have two different assets with the same expected value and same variance, but with zero correlation, then all mixtures of the two will have the same expected returns, but a 50/50 basket of the two assets will have 0.707 (1/sqrt(2)) times the variance of either of the two assets alone.
More generally, for a basket of non-correlated assets, a basket that maximizes expected returns divided by standard deviation of returns will never be 100% one asset. Returns are a linear function of the individual weights, but std. deviation of returns are a non-linear function. (This is true, regardless of statistical distribution of the random variables, as long as std. deviation is well-defined... no assumptions about normal distribution are involved.)
So, if you have two different assets with the same expected value and same variance, but with zero correlation, then all mixtures of the two will have the same expected returns, but a 50/50 basket of the two assets will have 0.707 (1/sqrt(2)) times the variance of either of the two assets alone.
More generally, for a basket of non-correlated assets, a basket that maximizes expected returns divided by standard deviation of returns will never be 100% one asset. Returns are a linear function of the individual weights, but std. deviation of returns are a non-linear function. (This is true, regardless of statistical distribution of the random variables, as long as std. deviation is well-defined... no assumptions about normal distribution are involved.)