Name the distribution

4. A spinner can land on the numbers \(10,12,14\) and 16 only and the probability of the spinner landing on each number is the same.
The random variable \(X\) represents the number that the spinner lands on when it is spun once.

1. State the name of the probability distribution of \(X\).
2. 1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find \(\operatorname { Var } ( X )\) A second spinner can land on the numbers 1, 2, 3, 4 and 5 only. The random variable \(Y\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(Y\) is given in the table below

      \(y\)	1	2	3	4	5
      \(\mathrm { P } ( Y = y )\)	\(\frac { 4 } { 30 }\)	\(\frac { 9 } { 30 }\)	\(\frac { 6 } { 30 }\)	\(\frac { 5 } { 30 }\)	\(\frac { 6 } { 30 }\)

3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\) The random variable \(W = a X + b\), where \(a\) and \(b\) are constants and \(a > 0\) Given that \(\mathrm { E } ( W ) = \mathrm { E } ( Y )\) and \(\operatorname { Var } ( W ) = \operatorname { Var } ( Y )\)
4. find the value of \(a\) and the value of \(b\). Each of the two spinners is spun once.
5. Find \(\mathrm { P } ( W = Y )\)

3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$

1. Write down the name given to this distribution. Find
2. \(\mathrm { P } ( X = 4 )\)
3. \(\mathrm { F } ( 3 )\)
4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
5. Write down \(\mathrm { E } ( X )\)
6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
8. find \(\operatorname { Var } ( a X - 3 )\)

3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$

1. Write down the name given to this distribution. Find
2. \(\mathrm { P } ( X = 4 )\)
3. \(\mathrm { F } ( 3 )\)
4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
5. Write down \(\mathrm { E } ( X )\)
6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
8. find \(\operatorname { Var } ( a X - 3 )\)
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1. The discrete random variable \(X\) has probability distribution

$$\mathrm { P } ( X = x ) = \frac { 1 } { 10 } \quad x = 1,2,3 , \ldots 10$$

1. Write down the name given to this distribution.
2. Write down the value of
   1. \(\mathrm { P } ( X = 10 )\)
   2. \(\mathrm { P } ( X < 10 )\) The continuous random variable \(Y\) has the normal distribution \(\mathrm { N } \left( 10,2 ^ { 2 } \right)\)
3. Write down the value of
   1. \(\mathrm { P } ( Y = 10 )\)
   2. \(\mathrm { P } ( Y < 10 )\)

2. The discrete random variable \(Q\) has the following probability distribution.

1. Write down the name of this distribution. The discrete random variable \(R\) has the following probability distribution.

   \(r\)	14	24	34	44	54
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)

2. State the relationship between \(R\) and \(Q\) in the form \(R = a Q + b\). Given that \(\mathrm { E } ( Q ) = 3\) and \(\operatorname { Var } ( Q ) = 2\),
3. find \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).

1. (a) (i) Name a suitable distribution for modelling the volume of liquid in bottles of wine sold as containing 75 cl .
   (ii) Explain why the mean in such a model would probably be greater than 75 cl .
   (b) (i) Name a suitable distribution for modelling the score on a single throw of a fair four-sided die with the numbers \(1,2,3\) and 4 on its faces.
   (ii) Use your suggested model to find the mean and variance of the score on a single throw of the die.
   (6 marks)
2. The events \(A\) and \(B\) are independent and such that

$$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) \text { and } \mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$$ (a) Show that \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
(b) Find \(\mathrm { P } ( A \cup B )\).
(c) Find \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).

6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:

1. State the name of this distribution.
2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:

   \(b\)	1	2	3
   \(\mathrm { P } ( B = b )\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)

3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
5. Find the probability distribution of \(C\).
6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).

5 A spinner has 8 equal areas numbered 1 to 8, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de9f0107-38de-4d0d-8391-4d29b98fa601-06_383_390_319_810} The spinner is spun and lands with one of its edges on the ground. 5

1. Assume that the spinner lands on each number with equal probability. 5
2. 1. State a distribution that could be used to model the number that the spinner lands on. 5
3. (ii) Use your distribution from part 5
4. 1. to find the probability that the spinner lands on a number greater than 5
      [0pt] [1 mark] 5
5. Clare spins the spinner 1000 times and records the results in the following table.

   Number landed on	1	2	3	4	5	6	7	8
   Frequency	37	64	112	161	308	156	109	53

   5
6. 1. Explain how the data shows that the model used in part (a) may not be valid.
      5
7. (ii) Describe how Clare's results could be used to adjust the model.