# Integrating area under a PDF distribution curve

I have plotted the PDF of a particular function using

Plot[PDF[NormalDistribution[64, 8.5333333], x], {x, 20, 100}]


This gives the curve

I want to find the probability of finding a data point in a particular region of this graph by integrating the area under the curve. I want to do this for a small section of the graph only, i.e. for a small range of values of $A_i$. How do I go about this?

• Try Quantile? – Mike Honeychurch Feb 16 '18 at 9:39

That's what CDF is good for!

f = PDF[NormalDistribution[64, 8.5333333]];
F = CDF[NormalDistribution[64, 8.5333333]];
F[b] - F[a]


returns the area of the graph of f between the points a and b.

• So if I want to find how many of my 8000 data points are in a region I just do (F[b]-F[a])*8000? – JJH Feb 16 '18 at 10:12
• Yepp........... – Henrik Schumacher Feb 16 '18 at 20:01

You can use the CDF function, e.g.

(* 5% one-tail probability *)
x = Quantile[NormalDistribution[0, 1], 0.05]


-1.64485

Show[Plot[PDF[NormalDistribution[0, 1], a], {a, -4, 4},
Ticks -> {{{-4, "μ - 4σ"}, {-2, "μ - 2σ"},
{2, "μ + 2σ"}, {4, "μ + 4σ"}}, None}],
Plot[PDF[NormalDistribution[0, 1], b], {b, -4, x},
PlotStyle -> None, Filling -> Axis]]


CDF[NormalDistribution[0, 1], x]


0.05

CDF returns 5%.