ps = c(1/2, 1/3, 1/6)
alphas = ps

rs = c(3, 2, 6)

alpha0 = 1 - sum(alphas)

E_ln_R1 = sum(ps * log(alpha0 + alphas * rs))

m = exp(E_ln_R1)

print(sprintf('E[ln(R_1)]= %f; m= %f', E_ln_R1, m))