Simple algebraic expression for P(X=x)

3 The random variable \(X\) takes the values \(- 2,1,2,3\). It is given that \(\mathrm { P } ( X = x ) = k x ^ { 2 }\), where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 The random variable \(X\) takes the values \(- 2,2\) and 3. It is given that $$\mathrm { P } ( X = x ) = k \left( x ^ { 2 } - 1 \right)$$ where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 The random variable \(X\) takes the values \(1,2,3,4\) only. The probability that \(X\) takes the value \(x\) is \(k x ( 5 - x )\), where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), in terms of \(k\).
2. Show that \(\operatorname { Var } ( X ) = 1.05\).

1 Becky sometimes works in an office and sometimes works at home. The random variable \(X\) denotes the number of days that she works at home in any given week. It is given that $$\mathrm { P } ( X = x ) = k x ( x + 1 )$$ where \(k\) is a constant and \(x = 1,2,3\) or 4 only.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

2 The random variable \(X\) takes the values \(- 2 , - 1,0,2,3\). It is given that \(\mathrm { P } ( X = x ) = k \left( x ^ { 2 } + 2 \right)\), where \(k\) is a positive constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find the value of \(\operatorname { Var } ( X )\).

3 A particular type of bird lays 1,2,3 or 4 eggs in a nest each year. The probability of \(x\) eggs is equal to \(k x\), where \(k\) is a constant.

1. Draw up a probability distribution table, in terms of \(k\), for the number of eggs laid in a year and find the value of \(k\).
2. Find the mean and variance of the number of eggs laid in a year by this type of bird.

3 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x ^ { 2 }\), where \(k\) is a constant and \(x\) takes values \(- 2 , - 1,2,4\) only.

1. Show that \(\mathrm { P } ( X = - 2 )\) has the same value as \(\mathrm { P } ( X = 2 )\).
2. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
3. Find \(\mathrm { E } ( X )\).

4 The random variable \(X\) takes the values \(- 1,1,2,3\) only. The probability that \(X\) takes the value \(x\) is \(k x ^ { 2 }\), where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

2 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x\), where \(x\) takes values \(1,2,3,4,5\) only.

1. Draw up a probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\).

3 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } - 1 \right) \text { for } r = 2,3,4,5 .$$

1. Show the probability distribution in a table, and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( r + 1 ) \quad \text { for } r = 1,2,3,4,5 .$$

1. Show that \(k = \frac { 1 } { 70 }\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4$$

1. Show that \(k = 0.05\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6$$

1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(k\)					\(11 k\)

2. Find the mean of \(X\). A fair six-sided die is rolled three times.
3. Find the probability that the total score is 16 .

5 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1. The random variable \(X\) has probability function

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

1. Show that \(k = 30\)
2. Find \(\mathrm { P } ( X \neq 4 )\)
3. Find the exact value of \(\mathrm { E } ( X )\)
4. Find the exact value of \(\operatorname { Var } ( X )\) Given that \(Y = 3 X - 1\)
5. find \(\mathrm { E } \left( Y ^ { 2 } \right)\)

2 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } - 1 \right) \text { for } r = 2,3,4,5 .$$

1. Show the probability distribution in a table, and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

2 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4 .$$

1. Show that \(k = 0.05\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = \frac { k } { r ( r - 1 ) } \text { for } r = 2,3,4,5,6 .$$

1. Show that the value of \(k\) is 1.2 . Using this value of \(k\), show the probability distribution of \(X\) in a table.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

12 A random variable \(X\) has probability distribution defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 1,2,3,4,5 , \\ 0 & \text { otherwise, } \end{cases}$$ where \(k\) is a constant.

1. Show that \(\mathrm { P } ( X = 3 ) = 0.2\).
2. Show in a table the values of \(X\) and their probabilities.
3. Two independent values of \(X\) are chosen, and their total \(T\) is found.
   1. Find \(\mathrm { P } ( T = 7 )\).
   2. Given that \(T = 7\), determine the probability that one of the values of \(X\) is 2 .

11 The discrete random variable \(X\) takes the values \(0,1,2,3,4\) and 5 with probabilities given by the formula $$\mathrm { P } ( X = x ) = k ( x + 1 ) ( 6 - x ) .$$

1. Find the value of \(k\). In one half-term Ben attends school on 40 days. The probability distribution above is used to model \(X\), the number of lessons per day in which Ben receives a gold star for excellent work.
2. Find the probability that Ben receives no gold stars on each of the first 3 days of the half-term and two gold stars on each of the next 2 days.
3. Find the expected number of days in the half-term on which Ben receives no gold stars.

4. The random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = k x , \quad x = 1,2 , \ldots , 5$$

1. Show that \(k = \frac { 1 } { 15 }\). Find
2. \(\mathrm { P } ( X < 4 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } ( 3 X - 4 )\).

1 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } + 3 r \right) \text { for } r = 1,2,3,4,5 \text {, where } k \text { is a constant. }$$

1. Complete the table below, using the copy in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)	\(4 k\)	\(10 k\)

2. Show that the value of \(k\) is 0.01 .
3. Draw a graph to illustrate the distribution.
4. Describe the shape of the distribution.
5. Find each of the following.

1 A fair five-sided spinner has sectors labelled 1, 2, 3, 4, 5. In a game at a stall at a charity event, the spinner is spun twice. The random variable \(X\) represents the lower of the two scores. The probability distribution of \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } ( 11 - 2 \mathrm { r } )\) for \(r = 1,2,3,4,5\),
where \(k\) is a constant.

1. Complete the copy of this table in the Printed Answer Booklet.

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)		\(7 k\)		\(3 k\)

2. Determine the value of \(k\).
3. Find each of the following.
   Given that the average profit that the stall-holder makes on one game is 25 pence, find the value of \(C\).

4 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \quad \text { for } r = 1,2,3,4,5,6 \text {, where } k \text { is a constant. }$$

1. Complete the table in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)

2. Show that the value of \(k\) is \(\frac { 1 } { 36 }\).
3. Draw a graph to illustrate the distribution.
4. In this question you must show detailed reasoning. Find
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\).
   A game consists of a player throwing two fair dice. The score is the maximum of the two values showing on the dice.
5. Show that the probability of a score of 3 is \(\frac { 5 } { 36 }\).
6. Show that the probability distribution for the score in the game is the same as the probability distribution of the random variable \(X\).
7. The game is played three times. Find
   • the mean of the total of the three scores.
   • the variance of the total of the three scores.