Note that this is the formula used to compute the $z$-score of a normal random variable! Is a standard normal random variable-it is normal with mean 0 and standard deviation 1. It turns out that if $X$ is a normal random variable with mean $\mu$ and standard deviation, then The trick is to "standardize" our normal random variables. This is clearly untenable, so we have to use some other tricks. We can then print up tables and tables of these approximations, and use them for calculations.īut there is a problem with this idea! The value of the integral will depend on the parameters $\mu$ and $\sigma$! This means that if we change these values even a little, then we have to compute an entirely new table. We can get approximates that are as good as we like (if we spend enough time and/or computer power on it), but all we'll ever have are approximations. The best that we can do is use numerical methods to approximate the values that this integral takes. It turns out that this integral does not have an antiderivative in terms of elementary functions-I'm not going to go into details about what this means, but long story short, it is very hard (possibly impossible) to compute this integral exactly. ![]() A normally distributed random variable $X$ has an associated probability distribution function (pdf) given by
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