# Quantiles of the Normal¶

Now take a Normal:

$X \sim N(\mu, \sigma^2),$

and let $$F(x) = p(X \le x)$$ be its CDF. The $$q$$-quantile of $$X$$ is defined through the solution of the non-linear equation:

$F(x_q) = q.$

Recall that we have managed to express the Normal CDF $$F(x)$$ in terms of the standard Normal CDF $$\Phi(z)$$:

$F(x) = \Phi\left(\frac{x-\mu}{\sigma}\right).$

We can use this to express $$x_q$$ in terms of $$z_q$$, i.e., the $$q$$-quantile of the standard Normal. We have by substitution in the defining equation of $$x_q$$:

$\Phi\left(\frac{x_q-\mu}{\sigma}\right) = q.$

By comparing to the defining equation of $$z_q$$,

$\Phi(z_q) = q,$

we get that:

$z_q = \frac{x_q - \mu}{\sigma},$

or, in terms of $$x_q$$:

$x_q = \mu + \sigma z_q.$

Alright, so if we know the $$q$$-quantiles of the standard Normal we can get the $$q$$-quantiles of a Normal. The median is trival:

$x_{0.5} = \mu,$

since $$z_{0.5}$$.

The $$0.025$$-quantile is:

$x_{0.025} \approx \mu - 1.96\sigma \approx \mu - 2\sigma.$

And the $$0.975$$-quantile is:

$x_{0.975} \approx \mu + 1.96\sigma \approx \mu + 2 \sigma.$

So, we can say (95% central credible interval):

$$X$$ is between $$\mu - 2\sigma$$ and $$\mu + 2\sigma$$ with 95% probability.

Similarly, we can find the $$0.001$$-quantile:

$x_{0.001} \approx \mu - 3.09 \sigma \approx = \mu - 3\sigma,$

and the $$0.999$$-quantile:

$x_{0.999} \approx \mu + 3.09\sigma \approx = \mu + 3 \sigma,$

which we can use to say (99.8% central credible interval):

$$X$$ is between $$\mu - 3\sigma$$ and $$\mu + 3 \sigma$$ with 99.8% probability.

These are good things to remember, but even if you don’t, you can still use scipy.stats to find quantile and central credible intervals of any Normal:

import scipy.stats as st
mu = 5
sigma = 2
X = st.norm(loc=mu, scale=sigma)
x_025 = X.ppf(0.025)
x_975 = X.ppf(0.975)
print('X is between {0:1.2f} and {1:1.2f} with probability 95%'.format(x_025, x_975))
print('Compare to the interval found through the standard normal')
print('** X is between {0:1.2f} and {1:1.2f} with probability 95%'.format(mu - sigma * 1.96,
mu + sigma * 1.96))

X is between 1.08 and 8.92 with probability 95%
Compare to the interval found through the standard normal
** X is between 1.08 and 8.92 with probability 95%


## Questions¶

• Modify the code above to find the 99.8% quantile of $$X$$.