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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!

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The question is formally well-defined, and Shai Covo has given an answer, but note that many distributions (these in particular) are not normal-the tails are much larger than Gauss would predict. Also, to know which is more likely to be negative, all you really need to know is how many standard deviations down from the mean zero is. You can compare the two numbers without calculating either one. –  Ross Millikan Dec 15 '10 at 14:06
    
@Ross Great points. Under the assumption that the two distributions have the same shape (i.e., are the same up to rescaling and translation), non-normality doesn't matter: it will affect the probability calculation but does not change the answer. –  whuber Dec 15 '10 at 17:19
    
This question was cross-posted at stats.stackexchange.com/q/5504/919 . –  whuber Dec 15 '10 at 17:21
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1 Answer

up vote 5 down vote accepted

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).

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Because the probability is a monotonic function of -mu/sigma, you don't have to compute it: you only need to compare the two values of -mu/sigma. –  whuber Dec 15 '10 at 17:18
    
Indeed, but it is useful to know how to compute it. –  Shai Covo Dec 15 '10 at 17:25
    
Thanks for the help! –  Matthew Dec 16 '10 at 9:06
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