For a continuous r

Next: Continuous Random Variables: Expectations and Variance
Previous: Continuous Random Variables: Basic Concepts

Percentiles and the Cumulative Distribution Function

For a continuous r.v. X, we only talk of the probability that X lies in some interval and not that X takes on a specific value. This is because P(X=x) = 0 for a continuous r.v.

It follows that P(X£ x) = P(X<x) and P(X³ x)=P(X>x) and P(a<X<b) = P(a£ X£ b) = P(a£ X<b) = P(a<X£ b).

The preceding is in general untrue for a discrete random variable.

The cumulative distribution function (cdf) for X is given by

Note that F(x) is a nondecreasing function that approaches 1 as x®¥ .

It follows that given two numbers a and b with a<b, P(a£X£b) = F(b) - F(a)

The n^th percentile of the distribution of X is that value on the x-axis for which n% of the area under the graph of f(x) lies to the left of the value, and (100-n)% of the area under the graph lies to the right of the value. The value is denoted by h(n/100). Note that P[X£h(n/100)] = n/100.

The median of X (denoted by

) is the 50^th percentile, i.e.,

=h(0.5) and P(X£)=F()=0.5

ENGR 0020 Home Page

Next: Continuous Random Variables: Expectations and Variance
Previous: Continuous Random Variables: Basic Concepts

Jayant Rajgopal