Derive marginal distribution of x and y derive conditional

Question

Q1. If a and b are fixed numbers and x and y are random variables, then show:

var(x)=E[x²] - [x]²

and

var(ax+by) = a² σ_x²+ b² σ_y² + 2abσ_xy

Hint: to prove the second one consider ax + by =z then var(z)=E[z²] - E[z]²

Q2. Use information provided in table below to:

i. Derive marginal distribution of x and y.

ii. Derive conditional distribution of y given x = 0 and x = 1 separately.

iii. Prove the "law of iterated expectations" i.e. E[y]=E[E[y¦x]]

Table1. Joint distribution of x and y

	X=0	X=1
Y=0	0.15	0.07
Y=1	0.15	0.63

Hint: E[E[y¦x] ]=E[y¦x=0]p(x=0)+E[y¦x=1]p(x=1).