We can estimate the probability of selecting a negative random
variable by evaluating the area under the curve in the region of
negative x. Approximating this area using a triangle will give us
an over-estimate of the actual probability.
Blue Curve: Pr(X< 0) \approx 0.5 x 2 x 0.25 = 0.25
Red Curve: Pr(X< 0) \approx 0.5 x 2 x 0.1 = 0.1
Black Curve: Pr(X< 0) \approx 0.5 x 2 x 0.05 = 0.05
A normal distribution is symmetric about its mean. This allows us
to estimate the mean of each distribution by inspection:
\mu_{Blue} = 1
\mu_{Red} = 2
\mu_{Black} = 3
We know that f(x) = \frac{1}{ \sigma \sqrt{2 \pi}}
e^{\frac{-(x-\mu)^2}{2 \sigma ^2}}
If we evaluate f(x) at x = \mu the exponential will disappear
(e^0 = 1)
We can then solve for \sigma
f(\mu) = \frac{1}{ \sigma \sqrt{2 \pi}}
\sigma =\frac{1}{\sqrt{2 \pi} f(\mu)}
Evaluating f(\mu) from the curves and substituting \mu into the
expression we find that: