Let X have the uniform distribution on the interval 0 1 Find

Let X have the uniform distribution on the interval [0, 1]. Find EX Repeat part a where X is uniform on the interval [a, b].

Solution

a)

Here,

f(x) = 1, 0<x<1
0, otherwise

Thus,

EX = Integral [x f(x) dx]

= Intergal [x dx]|(0,1)

= x^2/2|(0,1)

= 1/2 [ANSWER]

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b)

Here,

f(x) = 1/(b-a), a<x<b
0, otherwise

Thus,

EX = Integral [x f(x) dx]

= Intergal [x [1/(b-a)] dx]|(0,1)

=[1/(b-a)] x^2/2|(a,b)

= [1/(b-a)] [b^2 - a^2] /2

= [1/(b-a)] [(b+a)(b-a)] /2

= (b+a) /2 [ANSWER]