Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability distribution of the two measurements is a uniform distribution over the region 0 < x < 4, 0 < y, and x − 1< y < x + 1. That is, f_XY(x, y) = c for x and y in the region. Determine the value for c such that f_XY (x, y) is a joint probability density function.

Determine the following:

(a) P(X <0.5, Y <0.5) 

(b) P(X <0.5)

(c) E(X) 

(d) E(Y )

(e) Marginal probability distribution of X

(f) Conditional probability distribution of Y given X = 1

(g) E(Y | X = 1) 

(h) P(Y <0.5 | X = 1)