Normal distribution probability

Question

just a quick question dealing with probability. The annual returns on stocks and treasury bonds over the next 12 months are uncertain. Suppose that these returns can be described by normal distributions with stocks having a mean of 15% and a standard deviation of 20%, and bonds having a mean of 6% and a standard deviation of 9%. Which asset is more likely to have a negative return? Thanks for any help!

Answer 1

I don't know if that's what you are looking for, but if $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma > 0$, then $$ {\rm P}(X<0) = {\rm P}\bigg(\frac{{X - \mu }}{\sigma } < \frac{{0 - \mu }}{\sigma }\bigg) = {\rm P}\bigg(Z < \frac{{ - \mu }}{\sigma }\bigg), $$ where $Z$ is a standard normal random variable, i.e. having mean zero and unit variance (or standard deviation). The probability ${\rm P}(Z<z)$ can be found using a Normal Distribution Calculator, and many are available online (check, e.g., this one).