I recently came across this lower bound that I thought was interesting and not immediately obvious. Let and be the probability density function (PDF) and the cumulative distribution function (CDF) of the standard normal distribution, i.e.
Then the following inequality holds:
The plot below illustrates this inequality with the function on the LHS and RHS being in red and black respectively:
x <- seq(-30, 30) / 10 plot(x, x * pnorm(x) + dnorm(x), type = 'l', col = "red", ylab = "f(x)") lines(x, pmax(0, x), lty = 2)
The derivation of the inequality below is mine; there may be faster ways to do it. A key formula we will use in the proof is that for the derivative of : .
Let . First, we show that for all . Since, for all ,
the function is strictly increasing on the entire real line. Hence, using L’Hôpital’s rule,
Next, we show that for all . Note that we only have to prove this for , since for . Letting , we see that proving for is equivalent to proving for . Since
for all , the function is strictly decreasing for . Thus, for this range of values,
as required. (We used L’Hôpital’s rule again in the sequence of equalities above.)