Variance of linear transformation

1 The random variable \(W\) can take values 1,2 or 3 and has a discrete uniform distribution.

1. Write down the value of \(\mathrm { E } ( 2 W )\).
2. Find the value of \(\operatorname { Var } ( 2 W )\).
3. Determine the value of the constant \(k\) for which \(\mathrm { E } ( 2 \mathrm {~W} + \mathrm { k } ) = \operatorname { Var } ( 2 \mathrm {~W} + \mathrm { k } )\). The random variable \(S\) has the probability distribution shown in the following table.

   \(S\)	2	3	4	5	6
   \(P ( S = S )\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)

4. Calculate \(\operatorname { Var } ( S )\).

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

$$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } , \quad x = 1,2,3,4,5$$

1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 2\). Find
2. \(\mathrm { E } ( 3 X - 2 )\),
3. \(\operatorname { Var } ( 4 - 3 X )\).

4 A fair 8 -sided dice has faces labelled \(1,2 , \ldots , 8\). The random variable \(X\) represents the score when the dice is rolled once.

1. State the distribution of \(X\).
2. Find \(\mathrm { P } ( X < 4 )\).
3. Find each of the following.

4 The discrete random variable \(X\) follows a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\). The discrete random variable \(Y\) is defined by \(Y = 2 X\) 4

1. Using the standard results for \(\sum n , \sum n ^ { 2 }\) and \(\operatorname { Var } ( a X + b )\), prove that $$\operatorname { Var } ( Y ) = \frac { n ^ { 2 } - 1 } { 3 }$$ 4
2. A spinning toy can land on one of four values: 2, 4, 6 or 8
   Using a discrete uniform distribution, find the probability that the next value the toy lands on is greater than 2 4
3. State an assumption that is required for the discrete uniform distribution used in part (b) to be valid.

A pack of 52 cards contains 4 cards bearing each of the integers from 1 to 13. A card is selected at random. The random variable \(X\) represents the number on the card.

1. Find \(P(X \leq 5)\). [1 mark]
2. Name the distribution of \(X\) and find the expectation and variance of \(X\). [4 marks]

A hand of 12 cards consists of three 2s, four 3s, two 4s, two 5s and one 6. The random variable \(Y\) represents the number on a card chosen at random from this hand.

1. Draw up a table to show the probability distribution of \(Y\). [3 marks]
2. Calculate \(\text{Var}(3Y - 2)\). [6 marks]