If the random variable has a normal distribution with mean 0 and variance 1, we know that its tail probability is given by the integral

(that is, the area under a standard normal bell-shaped curve from up to ).

This integral does not have a closed form in terms of elementary functions (unless ). However we can find a good lower bound as

We show the proof of this result based on [1]. Interestingly it involves Jensen’s inequality: recall that this states that if is a convex function and denotes expectation with respect to some probability distribution, then for any random variable . Applied to the convex function this becomes

For fixed we now let have the distribution

Applying this in (2) gives

Evaluating these terms,

and

so (4) becomes

This is a quadratic inequality in being of the form

where and .

This is equivalent to , or

since the quantities involved are non-negative. In other words we have

which is equivalent to (1), as desired.

Another lower and upper bounds for the tail probability of a normal distribution are

a proof of which can be seen, e.g. in [2].

The bound we have looked at can be combined with the following upper bound to arrive at the following tighter bounds.

See [3] and [4] for a derivation of the upper bound as well as similar bounds.

#### References

[1] Z. W. Birnbaum, An Inequality for Mill’s Ratio, Ann. Math. Statist. Volume 13, Number 2 (1942), 245-246.

[2] J. D. Cook, Upper and lower bounds for the normal distribution function, 2009.

[4] L. Duembgen, Bounding Standard Gaussian Tail Probabilities, University of Bern Technical Report 76, 2010

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