Probabilities from a pdf, its properties and symmetry

Last time

Before we start: last time, in Probability density functions, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Area is probability

For a continuous variable, no single value has any probability. Instead the curve, the probability density function, tells you how much probability sits near each point, and the area under the curve between two values is the probability. Once you see that, three types of question appear: find an area, check a curve is legitimate, and use symmetry as a shortcut.

Two properties of a pdf

A probability density function f must never be negative. The curve is never below the x axis. And the total area under it must be exactly 1. To show a function could be a pdf, check both. Cambridge awards one mark for each.

Is it a pdf?

A straight line from 0, 0 to 2, 1. It is above the axis, and the area is the triangle: half times 2 times 1, which is 1. So it could be a pdf. A semicircle of radius r, above the axis. Its area is half pi r squared. Set that equal to 1: r squared is 2 over pi, so r is 0.798. Geometry saves integration. Use triangles, rectangles and semicircles when you can.

A past-paper example

May June 2023, Paper 61, question 2. The graph of f is a straight line from 0, 0 to 2, 1. Show that f could be a pdf. A semicircle centred at the origin is a pdf; find its radius. Part a: area half times 2 times 1 is 1, and f is not negative. Part b: half pi r squared equals 1 gives r equals root of 2 over pi, which is 0.798.