Functions of random variables

2 The independent discrete random variables \(X\) and \(Y\) can take the values 0,1 and 2 with probabilities as given in the tables. \(\quad\)

\(y\)	0	1	2
\(\mathrm { P } ( Y = y )\)	0.5	0.3	0.2

The random variables \(U\) and \(V\) are defined as follows: $$U = X Y , V = | X - Y | .$$

1. In the Printed Answer Book complete the table giving the joint distribution of \(U\) and \(V\).
2. Find \(\operatorname { Cov } ( U , V )\).
3. Find \(\mathrm { P } ( U V = 0 \mid V = 2 )\).

1. The discrete random variable \(X\) has probability distribution given by

The random variable \(Y = 2 - 5 X\)
Given that \(\mathrm { E } ( \mathrm { Y } ) = - 4\) and \(\mathrm { P } ( \mathrm { Y } \geqslant - 3 ) = 0.45\)

1. find the probability distribution of X . Given also that \(\mathrm { E } \left( \mathrm { Y } ^ { 2 } \right) = 75\)
2. find the exact value of \(\operatorname { Var } ( \mathrm { X } )\)
3. Find \(\mathrm { P } ( \mathrm { Y } > \mathrm { X } )\) \section*{Q uestion 2 continued}

1. The discrete random variable \(X\) has the following probability distribution.

1. Find \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) is defined in terms of the discrete random variable \(X\)
   When \(X\) is negative, \(Y = X ^ { 2 }\)
   When \(X\) is positive, \(Y = 3 X - 2\)
2. Find \(\mathrm { P } ( Y < 9 )\)
3. Find \(\mathrm { E } ( X Y )\)