Two unknowns from sum and expectation

7. The random variable \(X\) has probability distribution

1. Given that \(\mathrm { E } ( X ) = 4.5\), write down two equations involving \(p\) and \(q\). Find
2. the value of \(p\) and the value of \(q\),
3. \(\mathrm { P } ( 4 < X \leqslant 7 )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 27.4\), find
4. \(\operatorname { Var } ( X )\),
5. \(\mathrm { E } ( 19 - 4 X )\),
6. \(\operatorname { Var } ( 19 - 4 X )\).

3. The random variable \(X\) has probability distribution given in the table below. Given that \(\mathrm { E } ( X ) = 0.55\), find

1. the value of \(p\) and the value of \(q\),
2. \(\operatorname { Var } ( X )\),
3. \(\mathrm { E } ( 2 X - 4 )\).

6. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } a ( 3 - x ) & x = 0,1,2
b & x = 3 \end{array} \right.$$

1. Find \(\mathrm { P } ( X = 2 )\) and complete the table below.

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)	\(3 a\)	\(2 a\)		\(b\)

   Given that \(\mathrm { E } ( X ) = 1.6\)
2. Find the value of \(a\) and the value of \(b\). Find
3. \(\mathrm { P } ( 0.5 < X < 3 )\),
4. \(\mathrm { E } ( 3 X - 2 )\).
5. Show that the \(\operatorname { Var } ( X ) = 1.64\)
6. Calculate \(\operatorname { Var } ( 3 X - 2 )\).

1. A biased die with six faces is rolled. The discrete random variable \(X\) represents the score on the uppermost face. The probability distribution of \(X\) is shown in the table below.

1. Given that \(\mathrm { E } ( X ) = 4.2\) find the value of \(a\) and the value of \(b\).
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 20.4\)
3. Find \(\operatorname { Var } ( 5 - 3 X )\) A biased die with five faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The cumulative distribution function of \(Y\) is shown in the table below.

   \(y\)	1	2	3	4	5
   \(\mathrm {~F} ( y )\)	\(\frac { 1 } { 10 }\)	\(\frac { 2 } { 10 }\)	\(3 k\)	\(4 k\)	\(5 k\)

4. Find the value of \(k\).
5. Find the probability distribution of \(Y\). Each die is rolled once. The scores on the two dice are independent.
6. Find the probability that the sum of the two scores equals 2

2. The discrete random variable \(X\) has the following probability distribution, where \(p\) and \(q\) are constants.

1. Write down an equation in \(p\) and \(q\) Given that \(\mathrm { E } ( X ) = 0.4\)
2. find the value of \(q\)
3. hence find the value of \(p\) Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.275\)
4. find \(\operatorname { Var } ( X )\) Sarah and Rebecca play a game.
   A computer selects a single value of \(X\) using the probability distribution above.
   Sarah's score is given by the random variable \(S = X\) and Rebecca's score is given by the random variable \(R = \frac { 1 } { X }\)
5. Find \(\mathrm { E } ( R )\) Sarah and Rebecca work out their scores and the person with the higher score is the winner. If the scores are the same, the game is a draw.
6. Find the probability that
   1. Sarah is the winner,
   2. Rebecca is the winner.

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

1. Given that \(\mathrm { E } ( X ) = - 0.2\), find the value of \(\alpha\) and the value of \(\beta\).
2. Write down \(\mathrm { F } ( 0.8 )\).
3. Evaluate \(\operatorname { Var } ( X )\). Find the value of
4. \(\mathrm { E } ( 3 X - 2 )\),
5. \(\operatorname { Var } ( 2 X + 6 )\).

5. The discrete random variable \(X\) takes only the values 4, 5, 6, 7, 8 and 9. The probabilities of these values are given in the table: It is known that \(\mathrm { E } ( X ) = 6 \cdot 7\). Find

1. the values of \(p\) and \(q\),
2. the value of \(a\) for which \(\mathrm { E } ( 2 X + a ) = 0\),
3. \(\operatorname { Var } ( X )\). \section*{STATISTICS 1 (A) TEST PAPER 9 Page 2}

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

1. Find an expression for \(b\) in terms of \(a\).
2. Find an expression for \(\mathrm { E } ( X )\) in terms of \(a\). Given that \(\mathrm { E } ( X ) = \frac { 45 } { 16 }\),
3. find the values of \(a\) and \(b\),

3 A box contains a large number of pea pods. The number of peas in a pod may be modelled by the random variable \(X\). The probability distribution of \(X\) is tabulated below.

1. Two pods are picked randomly from the box. Find the probability that the number of peas in each pod is at most 4.
2. It is given that \(\mathrm { E } ( X ) = 5.1\).
   1. Determine the values of \(a\) and \(b\).
   2. Hence show that \(\operatorname { Var } ( X ) = 1.29\).
   3. Some children play a game with the pods, randomly picking a pod and scoring points depending on the number of peas in the pod. For each pod picked, the number of points scored, \(N\), is found by doubling the number of peas in the pod and then subtracting 5. Find the mean and the standard deviation of \(N\).
      [0pt] [3 marks]

3 The table shows the probability distribution of the random variable \(X\), where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { E } ( X ) = 1.8\), determine the values of \(a\) and \(b\). The random variable \(Y\) is given by \(Y = 10 - 3 X\).
2. Using the values of \(a\) and \(b\) which you found in part (a), find each of the following.
   • \(\mathrm { E } ( Y )\)
   • \(\operatorname { Var } ( Y )\)

1. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
   Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below.

Given that \(\mathrm { E } ( 4 N + 2 ) = 14.8\) and \(\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }\)

1. show that \(\operatorname { Var } ( N ) = 2.76\) The group decided to charge a 50p entry fee for the first photograph entered and then 20p for each extra photograph entered into the competition up to a maximum of \(\pounds 1\) per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £1 Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
2. calculate the expected entry fee per member. Bai suggests that, as the mean and variance are close, a Poisson distribution could be used to model the number of photographs entered by a member next year.
3. State a limitation of the Poisson distribution in this case.

4. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below. Given that \(\mathrm { E } ( 4 N + 2 ) = 14.8\) and \(\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }\)

1. show that \(\operatorname { Var } ( N ) = 2.76\) The group decided to charge a 50 p entry fee for the first photograph entered and then 20 p for each extra photograph entered into the competition up to a maximum of \(\pounds 1\) per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £l Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
2. calculate the expected entry fee per member.
   [0pt] [BLANK PAGE]

2.

1. A certain five-sided die is biased with faces numbered 0 to 4 . The score, Y , on each throw is a random variable with probability distribution given by:

   \(Y\)	0	1	2	3	4
   \(\mathrm { P } ( Y = y )\)	\(a\)	\(b\)	\(c\)	0.1	0.15

   where \(a\), \(b\) and \(c\) are constants. $$\begin{aligned} & \mathrm { P } ( Y = 1 ) = \mathrm { P } ( Y \geq 3 )
   & \mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 ) - 0.1 \end{aligned}$$ Find the values of \(a , b\) and \(c\).
   [0pt] [4 marks]
2. The same die is thrown 10 times. Find the probability that there are not more than 4 throws on which the score is 3 , stating the distribution used as well as any modelling assumptions made.
   [0pt] [4 marks]
3. A game uses the same biased die. The die is thrown once. If it shows 1, 3 or 4 then this number is the final score. If it shows 0 or 2 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
   (a) Find the probability that the final score is 3 .
   (b) Given that the die is thrown twice, find the probability that the final score is 3 .
   [0pt] [3 marks]
   [0pt] [BLANK PAGE]

4 A spinner has edges numbered \(1,2,3,4\) and 5 . When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table. It is known that \(\mathrm { E } ( N ) = 2.55\).

1. Find \(\operatorname { Var } ( N )\).
2. Find \(\mathrm { E } ( 3 N + 2 )\).
3. Find \(\operatorname { Var } ( 3 N + 2 )\).

2 In a fairground game a competitor scores \(0,1,2\) or 3 with probabilities given in the following table, where \(a\) and \(b\) are constants. The competitor's expected score is 0.9 .

1. Show that \(b = 0.15\).
2. Find the variance of the score.
3. The competitor has to pay \(\pounds 2.50\) to take part, and wins a prize of \(\pounds 2 X\), where \(X\) is the score achieved. Find the expectation of the competitor's loss.

16 The discrete random variable \(X\) has the probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c ( 7 - 2 x ) & x = 0,1,2,3
k & x = 4
0 & \text { otherwise } \end{array} \right.$$ where \(c\) and \(k\) are constants.
16

1. Show that \(16 c + k = 1\)
   16
2. Given that \(\mathrm { P } ( X \geq 3 ) = \frac { 5 } { 8 }\) find the value of \(c\) and the value of \(k\).