Find marginal pdfs and then compute other related values

Question

Probability and Statistical Inference (9th Edition) Chapter 4.4 Question 2

Answer 1

Part (a) $$f_x(x) = \int_0^1 (x+y)dy = x+{1\over2}$$ Now for the other one: $$f_y(y) = \int_0^1 (x+y)dx = y+{1\over2}$$From here, it's easy to see that multiplying them does not give us $f(x,y)$ $$f_x(x)*f_y(y)=(x+{1\over2})(y+{1\over2})\ne (x+y)$$ Part (b)

$$\mu X=\int_0^1x(x+{1\over2})dx = \mu Y$$ $$\mu^2X=\int_0^1x^2(x+{1\over2})dx = \mu^2 Y$$ $$\sigma^2X=\mu^2X-(\mu X)^2 = \sigma^2 Y$$