(Solved):   [15 marks] (b) (i) If X is a random variable, and a and bare fixed constants, show that E( ...

 

[15 marks] (b) (i) If X is a random variable, and a and bare fixed constants, show that E(aX+b) = a E(X)+b. You can use one of either the discrete case or continuous case to demonstate. (ii) A random variable X is defined as the 'winnings" on a scratchcard. It is known that E(X)=5 and E(X) = 132940. Find the variance of the variable X, 'winnings'. (iii) The scratchcard costs 5 euro to buy. Find the expected value and the variance of the variable 'gain or loss' of the scratchcard, i.e. X-5. (iv) A second scratchcard is launched that follows the same probability distribution but offers double the prizemoney. The scratchcard costs 10 euro to buy. Find the expected value and variance of the variable gain or loss' of the scratchcard, i.e. 2X - 10.