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A random variable X representing the time (in hours) a furnace is running is drawn from...

A random variable X representing the time (in hours) a furnace is running is drawn from Ω = [0, 4] according to the density function f(x) = Cx.

(a) What value must the constant C be in order for f to be a valid probability density function?

(b) Determine the cdf F(x) which gives P(X ≤ x) and use it to determine P(X ≤ 2).

(c) Determine the expected value of X

. (d) Determine E[6X + 1] and Var(6X + 1).

(e) Suppose the energy (in Joules) used by the furnace is a function of the time it is on. In particular, Energy Used = sin(X2 ) (please ignore this as it is not at all realistic).

Set up an integral that represents the expected value of the energy used but do not evaluate the integral.

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