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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 σx2+ b2 σy2 + 2abσxy
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
0.63
Hint: E[E[y¦x] ]=E[y¦x=0]p(x=0)+E[y¦x=1]p(x=1).
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