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Suppose that I have a normally distributed variable with unknown mean and known SD: $X \sim N(\mu, 1)$. I also know that $P(-2 < X < 2) = 0.3.$ Is it possible to calculate $\mu$ from this information alone?

As best I can tell, it isn't possible, because $\mu$ is not identified. But I am unsure of my thinking on this point. Normal Distribution - finding mean takes up a related question, but in that question, one has more information about X.

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No, it is (generally) not possible to unequivocally determine the mean from this information, since you could use either the upper or lower tail of the distribution to meet your requirements. If you find that offsetting the distribution to the right of your range by some amount yields the correct probability of falling in that range, offsetting the distribution the same amount to the left will yield an identical probability of falling in that range. In most cases, you will have two choices of mean that both meet your criteria, with no way to choose between them (except for the rare circumstances where the mean is 0).

In the example you give, the mean could either be 2.53 or -2.53, so we cannot determine a single choice of mean.

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  • $\begingroup$ Thank you. To follow up on your last paragraph: given the information that I provided in the post, is there a straightforward way to calculate that the answer must be either 2.53 or -2.53? I see that the two possibilities must be symmetric in this way, but I don't see a straightforward way to calculate them. $\endgroup$ – user697473 Mar 18 '20 at 14:55
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    $\begingroup$ @user697473 I imagine there is a way to calculate the mean, but I'm not exactly sure what it is - I just played around with the calculator at onlinestatbook.com/2/calculators/normal_dist.html to find a mean that fit. You could do this programmatically pretty easily by just trying a bunch of means to see what fits, but I'm not sure what the closed-form solution looks like. $\endgroup$ – Nuclear Hoagie Mar 18 '20 at 15:01

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