 Solution

For the uniform distribution on $[0,1]$, the mean is $1/2$ and the variance is $1/12$. The central limit theorem for independent identically distributed variables with finite nonzero variance givesChoose $\boxed{\alpha =1/2,\beta =1/2}$. ThenThe standard normal distribution function is continuous and strictly increasing, so its value at the finite positive number $\sqrt{12}$ is strictly between $1/2$ and one, as required.