Question

Suppose n numbers X1, X2, . . . , Xn are chosen from a uniform distribution on [0, 10]. We say that there is an increase at i if Xi < Xi+1. Let I be the number of increases. Find E[I].

Answer 1

The Uniform Distribution

This covers Uniform Distribution, Expectation and Variance, Proof of Expectation and Cumulative Distribution Function.

A continuous random variable X which has probability density function given by:

f(x) = 1 for a £ x £ b
         b - a

(and f(x) = 0 if x is not between a and b) follows a uniform distribution with parameters a and b. We write X ~ U(a,b)

1/(b-a)

Remember that the area under the graph of the random variable must be equal to 1 (see continuous random variables).

Expectation and Variance

If X ~ U(a,b), then:

• E(X) = ½ (a + b)

• Var(X) = (1/12)(b - a)²

Proof of Expectation

2

Cumulative Distribution Function

The cumulative distribution function can be found by integrating the p.d.f between 0 and t:

F(t) = 1-1 -dr - t-a