Chapter 10 Continuous probability distributions - Ugrad.math.ubc.ca

Math 103 Notes Chapter 10F(x) = π (− 6 ) ∣ ∣∣∣x12 π cos(π 6 s) = 1 (1 − cos( π )02 6 x) .This cumulative distribution function is shown in Figure 10.1(b).10.250.20.80.150.60.10.40.050.201 2 3 4 5 601 2 3 4 5 6xxFigure 10.1: (a) The probability density p(x), (left) and (b) the cumulative distribution F(x) (right)for example 1.10.3 Mean and medianRecall that we have defined the mean of a distribution of grades or mass in a previous chapter. Fora mass density ρ(x), the idea of the mean coincides with the center of mass of the distribution,¯x =∫ ba∫ baxρ(x) dx.ρ(x) dxThis definition can also be applied to a probability∫density, but in this case the integral in thebdenominator is simply 1 (by property 2), i.e. p(x) dx = 1. (The simplification is analogous toaan observation we made for expected value in a discrete probability distribution.)We define the mean of a probability density as follows:DefinitionFor a random variable in a ≤ x ≤ b and a probability density p(x) defined on this interval, themean or average value of x (also called the expected value), denoted ¯x is given by¯x =∫ baxp(x) dx.The idea of median encountered previously in grade distributions also has a parallel here. Simplyput, the median is the value of x that splits the probability distribution into two portions whoseareas are identical.v.2005.1 - January 5, 2009 4