Let a < b be real numbers. let x be a random variable such that it is known that a < = x < = b and for every subinterval of [a; b], the probability that x will belong to that subinterval is proportional to the length of that subinterval. then we say x has the uniform distribution on the interval [a; b] and the probability density function of x is given by: {1/b-a if a< =x< =b 0 otherwiseand we write x- u(a, b)(a) find e[x]. does this agree with your intuition? (b) show var[x]= b^2-a^2/12 hint: b^3- a^3 = (b- a) (b2^+ ab + a^2)(c) find the c. d.f of x.

a)

b)

So first we need to find the second central moment like this:

And now we can find the variance like this:

c)

Step-by-step explanation:

For this case we assume that

And the density function is given by

Part a

In order to find the expected value we need to do the following integral:

And yes agree with the intuition since is the middle value of the interval (a,b)

Part b

For this case we need to use the definition of variance:

So first we need to find the second central moment like this:

And now we can find the variance like this:

Part c

In order to find the cumulative distribution function we just ned to do the following integral:

step-by-step explanation: