5.02b Expectation and variance: discrete random variables

5. The random variable \(X\) represents the number on the uppermost face when a fair die is thrown.

1. Write down the name of the probability distribution of \(X\).
2. Calculate the mean and the variance of \(X\). Three fair dice are thrown and the numbers on the uppermost faces are recorded.
3. Find the probability that all three numbers are 6 .
4. Write down all the different ways of scoring a total of 16 when the three numbers are added together.
5. Find the probability of scoring a total of 16 .

6 A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for \(R\), the number of checkouts that are staffed at any given time, is $$\mathrm { P } ( R = r ) = \left\{ \begin{array} { c l } \frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { r - 1 } & r = 1,2,3 \\ k & r = 4 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 27 }\).
2. Find the probability that, at any given time, there will be at least 3 checkouts that are staffed.
3. It is suggested that the total number of customers, \(C\), that can be served at the checkouts per hour may be modelled by $$C = 27 R + 5$$ Find:
   1. \(\mathrm { E } ( C )\);
   2. the standard deviation of \(C\).

7 On a multiple choice examination paper, each question has five alternative answers given, only one of which is correct. For each question, candidates gain 4 marks for a correct answer but lose 1 mark for an incorrect answer.

1. James guesses the answer to each question.
   1. Copy and complete the following table for the probability distribution of \(X\), the number of marks obtained by James for each question.

      \(\boldsymbol { x }\)	4	- 1
      \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)

   2. Hence find \(\mathrm { E } ( X )\).
2. Karen is able to eliminate two of the incorrect answers from the five alternative answers given for each question before guessing the answer from those remaining. Given that the examination paper contains 24 questions, calculate Karen's expected total mark.

5 Joanne has 10 identically-shaped discs, of which 1 is blue, 2 are green, 3 are yellow and 4 are red. She places the 10 discs in a bag and asks her friend David to play a game by selecting, at random and without replacement, two discs from the bag.

1. Show that:
   1. the probability that the two discs selected are the same colour is \(\frac { 2 } { 9 }\);
   2. the probability that exactly one of the two discs selected is blue is \(\frac { 1 } { 5 }\).
2. Using the discs, Joanne plays the game with David, under the following conditions: If the two discs selected by David are the same colour, she will pay him 135p. If exactly one of the two discs selected by David is blue, she will pay him 145p. Otherwise David will pay Joanne 45p.
   1. When a game is played, \(X\) is the amount, in pence, won by David. Construct the probability distribution for \(X\), in the form of a table.
   2. Show that \(\mathrm { E } ( X ) = 33\).
3. Joanne modifies the game so that the amount per game, \(Y\) pence, that she wins may be modelled by $$Y = 104 - 3 X$$
   1. Determine how much Joanne would expect to win if the game is played 100 times.
   2. Calculate the standard deviation of \(Y\), giving your answer to the nearest 1 p .

4 The table below shows the probability distribution for the number of students, \(R\), attending classes for a particular mathematics module.

1. Find values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
2. The number of students, \(S\), attending classes for a different mathematics module is such that $$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$ Find values for the mean and variance of:
   1. \(T = R + S\);
   2. \(\quad D = S - R\).

6 The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).

1. Prove that \(\mathrm { E } ( X ) = \lambda\).
2. By first proving that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), or otherwise, prove that \(\operatorname { Var } ( X ) = \lambda\).

6

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove that \(\mathrm { E } ( X ) = n p\).
   2. Given that \(\mathrm { E } \left( X ^ { 2 } \right) - \mathrm { E } ( X ) = n ( n - 1 ) p ^ { 2 }\), show that \(\operatorname { Var } ( X ) = n p ( 1 - p )\).
   3. Given that \(X\) is found to have a mean of 3 and a variance of 2.97, find values for \(n\) and \(p\).
   4. Hence use a distributional approximation to estimate \(\mathrm { P } ( X > 2 )\).
2. Dressher is a nationwide chain of stores selling women's clothes. It claims that the probability that a customer who buys clothes from its stores uses a Dressher store card is 0.45 . Assuming this claim to be correct, use a distributional approximation to estimate the probability that, in a random sample of 500 customers who buy clothes from Dressher stores, at least half of them use a Dressher store card.

3 The discrete random variable \(X\) has the following probability distribution The continuous random variable \(Y\) has the following probability density function $$\mathrm { f } ( y ) = \begin{cases} \frac { 1 } { 64 } y ^ { 3 } & 0 \leq y \leq 4 \\ 0 & \text { otherwise } \end{cases}$$ Given that \(X\) and \(Y\) are independent, show that \(\mathrm { E } \left( X ^ { 2 } + Y ^ { 2 } \right) = \frac { 1327 } { 60 }\)

6 The discrete random variable \(Y\) has the probability function $$\mathrm { P } ( Y = y ) = \begin{cases} 2 k y & y = 1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. Show that \(\operatorname { Var } ( 5 Y - 2 ) = 25\) \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-07_2488_1716_219_153}

5 The discrete random variable \(X\) has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 1 } { n } & x = 1,2 , \ldots , n \\ 0 & \text { otherwise } \end{cases}$$ 5

1. 1. Prove that \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\) [0pt] [3 marks]
      5
      1. (ii) Prove that \(\operatorname { Var } ( X ) = \frac { n ^ { 2 } - 1 } { 12 }\)

         5	State two conditions under which a discrete uniform distribution can be used to model the score when a cubic dice is rolled. [2 marks]

3 The discrete random variable \(A\) has the following probability distribution function $$\mathrm { P } ( A = a ) = \begin{cases} 0.45 & a = 0 \\ 0.25 & a = 1 \\ 0.3 & a = 2 \\ 0 & \text { otherwise } \end{cases}$$ 3

1. Find the median of \(A\) 3
2. Find the standard deviation of \(A\), giving your answer to three significant figures.
   3
3. \(\quad\) Find \(\operatorname { Var } ( 9 A - 2 )\)

3 The discrete random variable \(X\) has probability distribution Show that \(\mathrm { E } ( 5 X - 7 ) = 3.5\)

4 The discrete random variable \(Y\) has probability distribution The standard deviation of \(Y\) is \(s\) 4

1. Show that \(s = 10.53\) correct to two decimal places.
   [0pt] [4 marks]
   4
2. The median of \(Y\) is \(m\) Find \(\mathrm { P } ( Y > m - 1.5 s )\)

3 In a game, it is only possible to score 10, 20 or 30 points. The probability of scoring 20 points is twice the probability of scoring 30 points.
The probability of scoring 20 points is half the probability of scoring 10 points.
3

1. Find the mean points scored when the game is played once, giving your answer to two decimal places.
   3
2. Mina plays the game.
   Her father, Michael, tells her that he will multiply her score by 5 and then subtract 10 He will then give her the value he has calculated in pence rounded to the nearest penny. Calculate the expected value in pence that Mina receives.

5 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x \leq 1 \\ \frac { 1 } { 10 } x - \frac { 1 } { 10 } & 1 < x \leq 6 \\ \frac { 1 } { 90 } x ^ { 2 } + \frac { 1 } { 10 } & 6 < x \leq 9 \\ 1 & x > 9 \end{array} \right.$$ 5

1. Find the probability density function \(\mathrm { f } ( x )\) 5
2. Show that \(\operatorname { Var } ( X ) = \frac { 6737 } { 1200 }\) \includegraphics[max width=\textwidth, alt={}, center]{3ef4c3fd-cbf0-4ac0-a072-a07d763fd50a-07_2488_1716_219_153}

7 The random variable \(X\) has an exponential distribution with parameter \(\lambda\) 7

1. Prove that \(\mathrm { E } ( X ) = \frac { 1 } { \lambda }\) 7
2. Prove that \(\operatorname { Var } ( X ) = \frac { 1 } { \lambda ^ { 2 } }\)

6 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 0 \\ b & x = 1 \\ c & x = 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The mean of \(X\) is 1.2 and the variance of \(X\) is 0.56
6

1. Deduce the values of \(a , b\) and \(c\) 6
2. The continuous random variable \(Y\) is independent of \(X\) and has variance 15 Find \(\operatorname { Var } ( X - 2 Y - 11 )\) [0pt] [2 marks]

8 The continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \begin{cases} 0 & x = 0 \\ \mathrm { e } ^ { k x } - 1 & 0 \leq x \leq 5 \\ 1 & x > 5 \end{cases}$$ 8

1. Show that \(k = \frac { 1 } { 5 } \ln 2\) [0pt] [2 marks]
   8
2. Show that the median of \(X\) is \(a \frac { \ln b } { \ln 2 } - c\), where \(a , b\) and \(c\) are integers to be found.
   

   8	Show that the mean of \(X\) is \(p - \frac { q } { \ln 2 }\), where \(p\) and \(q\) are integers to be found.

6 A game consists of two rounds. The first round of the game uses a random number generator to output the score \(X\), a real number between 0 and 10 6

1. Find \(\mathrm { P } ( X > 4 )\) 6
2. The second round of the game uses an unbiased dice, with faces numbered 1 to 6 , to give the score \(Y\) The variables \(X\) and \(Y\) are independent.
   6 (b) (i) Find the mean total score of the game.
   6 (b) (ii) Find the variance of the total score of the game.

8 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k \sin 2 x & 0 \leq x \leq \frac { \pi } { 6 } \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. 8

1. Show that \(k = 4\) 8
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) 8
3. Find the median of \(X\), giving your answer to three significant figures. 8
4. Find the mean of \(X\) giving your answer in the form \(\frac { 1 } { a } ( b \sqrt { 3 } - \pi )\) where \(a\) and \(b\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-14_2492_1721_217_150}

5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\)

1 In a promotion for a new type of cereal, a toy dinosaur is included in each pack. There are three different types of dinosaur to collect. They are distributed, with equal probability, randomly and independently in the packs. Sam is trying to collect all three of the dinosaurs.

1. Find the probability that Sam has to open only 3 packs in order to collect all three dinosaurs. Sam continues to open packs until she has collected all three dinosaurs, but once she has opened 6 packs she gives up even if she has not found all three. The random variable \(X\) represents the number of packs which Sam opens.
2. Complete the table below, using the copy in the Printed Answer Booklet, to show the probability distribution of \(X\).

   \(r\)	3	4	5	6
   \(\mathrm { P } ( X = r )\)		\(\frac { 2 } { 9 }\)	\(\frac { 14 } { 81 }\)

   \section*{(iii) In this question you must show detailed reasoning.} Find

11 Two girls, Lili and Hui, play a game with a fair six-sided dice. The dice is thrown 10 times. \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) represent the scores on the \(1 ^ { \text {st } } , 2 ^ { \text {nd } } , \ldots , 10 ^ { \text {th } }\) throws of the dice. \(L\) denotes Lili's score and \(L = 10 X _ { 1 }\). \(H\) denotes Hui's score and \(H = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).

1. Calculate
   The spreadsheet below shows a simulation of 25 plays of the game. The cell E3, highlighted, shows the score when the dice is thrown the fourth time in the first game. \begin{table}[h]

   	A	B	C	D	E	F	G	H	I	J	K	L	M	N
   1	Throw of dice												Lili's	Hui's
   2		1	2	3	4	5	6	7	8	9	10		score	score
   3	Game 1	3	5	2	1	1	3	1	1	1	4		30	22
   4	Game 2	6	3	2	4	4	3	5	3	3	5		60	38
   5	Game 3	6	4	2	6	5	2	1	5	2	3		60	36
   6	Game 4	1	5	1	6	6	3	1	4	6	2		10	35
   7	Game 5	4	4	3	1	6	4	4	1	6	2		40	35
   8	Game 6	2	1	5	1	2	5	1	5	2	3		20	27
   9	Game 7	1	1	3	4	4	5	6	3	4	2		10	33
   10	Game 8	1	1	3	6	3	4	4	5	2	3		10	32
   11	Game 9	2	2	2	4	3	2	1	5	5	6		20	32
   12	Game 10	3	5	3	3	5	3	4	3	1	1		30	31
   13	Game 11	5	3	6	5	5	4	2	1	1	5		50	37
   14	Game 12	6	4	3	2	4	1	3	3	5	3		60	34
   15	Game 13	2	3	2	1	2	2	2	2	2	1		20	19
   16	Game 14	4	1	3	3	1	2	6	6	1	3		40	30
   17	Game 15	5	1	2	6	3	4	6	3	6	4		50	40
   18	Game 16	3	6	1	1	5	3	1	3	3	3		30	29
   19	Game 17	5	2	5	2	4	5	2	2	3	4		50	34
   20	Game 18	3	6	3	5	5	2	3	1	1	2		30	31
   21	Game 19	6	6	3	1	5	6	3	4	1	6		60	41
   22	Game 20	2	6	4	5	6	5	2	4	3	3		20	40
   23	Game 21	5	3	5	4	5	3	3	6	6	1		50	41
   24	Game 22	6	3	5	5	6	3	5	6	1	1		60	41
   25	Game 23	5	4	5	5	6	4	2	1	3	6		50	41
   26	Game 24	3	5	2	3	2	4	3	2	3	3		30	30
   27	Game 25	5	2	4	2	4	5	2	2	5	2		50	33
   28
   29												mean	37.60	33.68
   30												sd	17.39	5.77

   \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{table}
2. Use the simulation to estimate \(\mathrm { P } ( L > 40 )\) and \(\mathrm { P } ( H > 40 )\).
3. (A) Calculate the exact value of \(\mathrm { P } ( L > 40 )\).
   (B) Comment on how the exact value compares with your estimate of \(\mathrm { P } ( L > 40 )\) in part (v). Hui wonders whether it is appropriate to use the Central Limit Theorem to approximate the distribution of \(X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 }\).
4. (A) State what type of diagram Hui could draw, based on the output from the spreadsheet, to investigate this.
   (B) Explain how she should interpret the diagram.
5. (A) Calculate an approximate value of \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { 10 } > 40 \right)\) using the Central Limit Theorem.
   (B) Comment on how this value compares with your estimate of \(\mathrm { P } ( H > 40 )\) in part (v). \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
   For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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1 The probability distribution for the discrete random variable \(W\) is given in the table.

1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
2. Find \(\operatorname { Var } ( 3 W + 6 )\).

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

1. Find the value of the constant \(k\)
2. Find the probability distribution of \(X\) A biased die with eight faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The probability distribution of \(Y\) is shown in the table below, where \(a\) and \(b\) are constants.

   \(y\)	1	2	3	4	5	6	7	8
   \(\mathrm { P } ( Y = y )\)	\(a\)	\(a\)	\(a\)	\(b\)	\(b\)	\(b\)	0.11	0.05

   Given that \(\mathrm { E } ( Y ) = 4.02\)
3. form and solve two equations in \(a\) and \(b\) to show that \(a = 0.15\) You must show your working.
   (Solutions relying on calculator technology are not acceptable.)
4. Show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 20.7\)
5. Find \(\operatorname { Var } ( 5 - 2 Y )\) These dice are each rolled once. The scores on the two dice are independent.
6. Find the probability that the sum of these two scores is 3