Simple algebraic expression for P(X=x)

5. The probability function of a discrete random variable \(X\) is given by $$\mathrm { p } ( x ) = k x ^ { 2 } \quad x = 1,2,3$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 1 } { 14 }\) Find
2. \(\mathrm { P } ( X \geqslant 2 )\)
3. \(\mathrm { E } ( X )\)
4. \(\operatorname { Var } ( 1 - X )\)

3. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k x , & x = 1,2,3,4,5 ,
0 , & \text { otherwise } . \end{cases}\)

1. Show that \(k = \frac { 1 } { 15 }\). Find the value of
2. \(\mathrm { E } ( 2 X + 3 )\),
3. \(\operatorname { Var } ( 2 X - 4 )\).
   (6)

4. The discrete random variable \(X\) has probability function \(\mathrm { P } ( X = x ) = k ( x + 4 )\). Given that \(X\) can take any of the values \(- 3 , - 2 , - 1,0,1,2,3,4\),

1. find the value of the constant \(k\).
2. Find \(\mathrm { P } ( X < 0 )\).
3. Show that the cumulative distribution \(\mathrm { F } ( x )\) is given by $$\mathrm { F } ( x ) = \lambda ( x + 4 ) ( x + 5 )$$ where \(\lambda\) is a constant to be found. \section*{STATISTICS 1 (A) TEST PAPER 4 Page 2}

5. The discrete random variable \(X\) has the probability function shown below. $$P ( X = x ) = \left\{ \begin{array} { c c } k x , & x = 2,3,4,5,6
0 , & \text { otherwise } \end{array} \right.$$

1. Find the value of \(k\).
2. Show that \(\mathrm { E } ( X ) = \frac { 9 } { 2 }\). Find
3. \(\mathrm { P } [ X > \mathrm { E } ( X ) ]\),
4. \(\mathrm { E } ( 2 X - 5 )\),
5. \(\operatorname { Var } ( X )\).

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.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)

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.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • The stall-holder charges a player \(C\) pence to play the game, and then pays the player \(50 X\) pence, where \(X\) is the player's score.
   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.

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

1. Show that \(k = \frac { 1 } { 35 }\). The distribution of \(X\) is shown in the table.

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 35 }\)	\(\frac { 2 } { 35 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 17 } { 35 }\)

2. Comment briefly on the shape of the distribution.
3. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   The random variable \(Y\) is given by \(Y = 5 X - 10\).
4. Find each of the following.
   • \(\mathrm { E } ( Y )\)
   • \(\operatorname { Var } ( Y )\)

1 A fair six-sided dice is rolled three times.
The random variable \(X\) represents the lowest of the three scores.
The probability distribution of \(X\) is given by the formula
\(\mathrm { P } ( X = r ) = k \left( 127 - 39 r + 3 r ^ { 2 } \right)\) for \(r = 1,2,3,4,5,6\).

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

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(91 k\)	\(61 k\)	\(37 k\)

2. Show that \(k = \frac { 1 } { 216 }\).
3. Draw a graph to illustrate the distribution.
4. Comment briefly on the shape of the distribution.
5. In this question you must show detailed reasoning. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)

8. The discrete random variable \(R\) takes even integer values from 2 to \(2 n\) inclusive.
The probability distribution of \(R\) is given by $$\mathrm { P } ( R = r ) = \frac { r } { k } \quad r = 2,4,6 , \ldots , 2 n$$ where \(k\) is a constant.

1. Show that \(k = n ( n + 1 )\) When \(n = 20\)
2. find the exact value of \(\mathrm { P } ( 16 \leqslant R < 26 )\) When \(n = 20\), a random value \(g\) of \(R\) is taken and the quadratic equation in \(x\) $$x ^ { 2 } + g x + 3 g = 5$$ is formed.
3. Find the exact probability that the equation has no real roots.

14 A probability distribution is given by $$\mathrm { P } ( X = x ) = c ( 4 - x ) , \text { for } x = 0,1,2,3$$ where \(c\) is a constant.
14

1. Show that \(c = \frac { 1 } { 10 }\)
   14
2. Calculate \(\mathrm { P } ( X \geq 1 )\)