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].
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)
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
Cumulative Distribution Function
The cumulative distribution function can be found by integrating the p.d.f between 0 and t:
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