5.02b Expectation and variance: discrete random variables

1. A random sample of 75 packets of seeds is selected from a production line. Each packet contains 12 seeds. The seeds are planted and the number of seeds that germinate from each packet is recorded. The results are as follows.

1. Show that the probability of a randomly selected seed from this sample germinating is 0.82 A gardener suggests that a binomial distribution can be used to model the number of seeds that germinate from a packet of 12 seeds. She uses a binomial distribution with the estimated probability 0.82 of a seed germinating. Some of the calculated expected frequencies are shown in the table below.

   Number of seeds that germinate from each packet	6 or fewer	7	8	9	10	11	12
   Expected frequency	\(s\)	2.80	7.97	\(r\)	22.04	18.26	6.93

2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
3. Test, at the \(10 \%\) level of significance, whether or not these data suggest that the binomial distribution is a suitable model for the number of seeds that germinate from a packet of 12 seeds. State your hypotheses clearly and show your working.

1. A researcher is looking into the effectiveness of a new medicine for the relief of symptoms. He collects random samples of 8 people who are taking the medicine from each of 50 different medical practices. The number of people who say that the medicine is a success, in each sample, is recorded. The results are summarised in the table below.

The researcher decides to model this data using a binomial distribution.

1. State two necessary assumptions that the researcher made in order to use this model.
2. Show that the mean number of successes per sample is 3.54 He decides to use this mean to calculate expected frequencies. The results are shown in the table below.

   Number of successes	0	1	2	3	4	5	6	7	8
   Expected frequency	0.47	2.96	8.23	13.07	\(f\)	8.23	3.27	0.74	\(g\)

3. Calculate the value of \(f\) and the value of \(g\). Give your answers to 2 decimal places.
4. Stating your hypotheses clearly, test at the \(10 \%\) level of significance, whether or not the binomial distribution is a suitable model for the number of successes in samples of 8 people.

6. A total of 100 random samples of 6 items are selected from a production line in a factory and the number of defective items in each sample is recorded. The results are summarised in the table below.

1. Show that the mean number of defective items per sample is 2.91 A factory manager suggests that the data can be modelled by a binomial distribution with \(n = 6\). He uses the mean from the sample above and calculates expected frequencies as shown in the table below.

   Number of defective items	0	1	2	3	4	5	6
   Expected frequency	1.87	10.54	24.82	\(a\)	22.01	8.29	\(b\)

2. Calculate the value of \(a\) and the value of \(b\) giving your answers to 2 decimal places.
3. Test, at the \(5 \%\) level, whether or not the binomial distribution is a suitable model for the number of defective items in samples of 6 items. State your hypotheses clearly.

3. The random variable \(X\) has the discrete uniform distribution over the set of consecutive integers \(\{ - 7 , - 6 , \ldots , 10 \}\).
Calculate (a) the expectation and variance of \(X\),
(b) \(\mathrm { P } ( X > 7 )\),
(c) the value of \(n\) for which \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).

6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables: The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).

1. Tabulate the probability distribution for \(Z\).
2. Calculate \(\mathrm { E } ( Z )\).
3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
4. Calculate Var (3Z-4).

1. Two spinners are in the form of an equilateral triangle, whose three regions are labelled 1,2 and 3, and a square, whose four regions are labelled \(1,2,3\) and 4 . Both spinners are biased and the probability distributions for the scores \(X\) and \(Y\) obtained when they are spun are respectively:

1. Find the values of \(p\) and \(q\).
2. Find the probability that, when the two spinners are spun together, the sum of the two scores is (i) 5, (ii) less than 4 .
3. State an assumption that you have made in answering part (b) and explain why it is likely to be justifiable.
4. Calculate \(\mathrm { E } ( X + Y )\).
   

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

1. \(k\),
2. \(\mathrm { F } ( 2 )\),
3. \(\mathrm { P } ( 1.3 < X \leq 3.8 )\),
4. \(\mathrm { E } ( X )\),
5. \(\operatorname { Var } ( 3 X + 2 )\).

7. A bag contains 4 red and 2 blue balls, all of the same size. A ball is selected at random and removed from the bag. This is repeated until a blue ball is pulled out of the bag. The random variable \(B\) is the number of balls that have been removed from the bag.

1. Show that \(\mathrm { P } ( B = 2 ) = \frac { 4 } { 15 }\).
2. Find the probability distribution of \(B\).
3. Find \(\mathrm { E } ( B )\). The bag and the same 6 balls are used in a game at a funfair. One ball is removed from the bag at a time and a contestant wins 50 pence if one of the first two balls picked out is blue.
4. What are the expected winnings from playing this game once? For \(\pounds 1\), a contestant gets to play the game three times.
5. What is the expected profit or loss from the three games?

5. The discrete random variable \(Y\) has the following cumulative distribution function.

1. Write down the probability distribution of \(Y\).
2. Find \(\mathrm { P } ( 1 \leq Y < 3 )\).
3. Show that \(\mathrm { E } ( Y ) = 2.7\)
4. Find \(\mathrm { E } ( 2 Y + 4 )\).
5. Find \(\operatorname { Var } ( Y )\).

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 \(b\),

1. The discrete random variable \(Y\) has the following probability distribution.

Find

1. \(\mathrm { F } ( 0.5 )\),
2. \(\mathrm { P } \left( { } ^ { - } 1 < Y < 1.9 \right)\),
3. \(\mathrm { E } ( Y )\),
4. \(\mathrm { E } ( 3 Y - 1 )\).
   

5. A group of children were each asked to try and complete a task to test hand-eye coordination. Each child repeated the task until he or she had been successful or had made four attempts. The number of attempts made by the children in the group are summarised in the table below.

1. Calculate the mean and standard deviation of the number of attempts made by each child. It is suggested that the number of attempts made by each child could be modelled by a discrete random variable \(X\) with the probability function $$P ( X = x ) = \left\{ \begin{array} { c c } k \left( 20 - x ^ { 2 } \right) , & x = 1,2,3,4 \\ 0 , & \text { otherwise } \end{array} \right.$$
2. Show that \(k = \frac { 1 } { 50 }\).
3. Find \(\mathrm { E } ( X )\).
4. Comment on the suitability of this model.

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

1. Find and simplify an expression in terms of \(k\) for \(\mathrm { E } ( X )\). Given that \(\mathrm { E } ( X ) = 9\),
2. find the value of \(k\).
   

2. The discrete random variable \(X\) has the probability function shown below. $$P ( X = x ) = \left\{ \begin{array} { l c } \frac { k } { x } , & x = 1,2,3,4 \\ 0 , & \text { otherwise } . \end{array} \right.$$

1. Show that \(k = \frac { 12 } { 25 }\) Find
2. \(\mathrm { F } ( 2 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } \left( X ^ { 2 } + 2 \right)\).

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 \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).

3. The random variable \(X\) is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of \(a\) and \(b\) for

1. \(\mathrm { E } ( 2 X + 3 )\),
2. \(\quad \operatorname { Var } ( 2 X + 3 )\),
3. \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$

5 The Globe Express agency organises trips to the theatre. The cost, \(\pounds X\), of these trips can be modelled by the following probability distribution:

1. Calculate the mean and standard deviation of \(X\).
2. For special celebrity charity performances, Globe Express increases the cost of the trips to \(\pounds Y\), where $$Y = 10 X + 250$$ Determine the mean and standard deviation of \(Y\).

4 The number of fish, \(X\), caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution:

1. Find the value of \(k\).
2. Find:
   1. \(\mathrm { E } ( X )\);
   2. \(\operatorname { Var } ( X )\).
3. When Pearl sells her fish, she earns a profit, in pounds, given by $$Y = 5 X + 2$$ Find:
   1. \(\mathrm { E } ( Y )\);
   2. the standard deviation of \(Y\).

5 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { x } { 20 } & x = 1,2,3,4,5 \\ \frac { x } { 24 } & x = 6 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X \geqslant 5 )\).
2. 1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 24 }\).
   2. Hence, or otherwise, show that \(\operatorname { Var } \left( \frac { 1 } { X } \right) = 0.036\), correct to three decimal places.
3. Calculate the mean and the variance of \(A\), the area of rectangles having sides of length \(X + 3\) and \(\frac { 1 } { X }\).

6

1. Ali has a bag of 10 balls, of which 5 are red and 5 are blue. He asks Ben to select 5 of these balls from the bag at random. The probability distribution of \(X\), the number of red balls that Ben selects, is given in Table 1. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 1}

   \(\boldsymbol { x }\)	0	1	2	3	4	5
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	\(\frac { 1 } { 252 }\)	\(\frac { 25 } { 252 }\)	\(\frac { 100 } { 252 }\)	\(a\)	\(\frac { 25 } { 252 }\)	\(\frac { 1 } { 252 }\)

   \end{table}
   1. State the value of \(a\).
   2. Hence write down the value of \(\mathrm { E } ( X )\).
   3. Determine the standard deviation of \(X\).
2. Ali decides to play a game with Joanne using the same 10 balls. Joanne is asked to select 2 balls from the bag at random. Ali agrees to pay Joanne 90 p if the two balls that she selects are the same colour, but nothing if they are different colours. Joanne pays 50 p to play the game. The probability distribution of \(Y\), the number of red balls that Joanne selects, is given in Table 2. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 2}

   \(\boldsymbol { y }\)	0	1	2
   \(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	\(\frac { 2 } { 9 }\)	\(\frac { 5 } { 9 }\)	\(\frac { 2 } { 9 }\)

   \end{table}
   1. Determine whether Joanne can expect to make a profit or a loss from playing the game once.
   2. Hence calculate the expected size of this profit or loss after Joanne has played the game 100 times.
      (3 marks)

4

1. A red biased tetrahedral die is rolled. The number, \(X\), on the face on which it lands has the probability distribution given by

   \(\boldsymbol { x }\)	1	2	3	4
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.2	0.1	0.4	0.3

   1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   2. The red die is now rolled three times. The random variable \(S\) is the sum of the three numbers obtained. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
2. A blue biased tetrahedral die is rolled. The number, \(Y\), on the face on which it lands has the probability distribution given by $$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3 \\ \frac { 7 } { 10 } & y = 4 \end{cases}$$ The random variable \(T\) is the value obtained when the number on the face on which it lands is multiplied by 3 . Calculate \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
3. Calculate:
   1. \(\mathrm { P } ( X > 1 )\);
   2. \(\mathrm { P } ( X + T \leqslant 9\) and \(X > 1 )\);
   3. \(\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )\).

5

1. Joshua plays a game in which he repeatedly tosses an unbiased coin. His game concludes when he obtains either a head or 5 tails in succession. The random variable \(N\) denotes the number of tosses of his coin required to conclude a game. By completing Table 3 below, calculate \(\mathrm { E } ( N )\).
2. Joshua's sister, Ruth, plays a separate game in which she repeatedly tosses a coin that is biased in such a way that the probability of a head in a single toss of her coin is \(\frac { 1 } { 4 }\). Her game also concludes when she obtains either a head or 5 tails in succession. The random variable \(M\) denotes the number of tosses of her coin required to conclude her game. Complete Table 4 below.
3. 1. Joshua and Ruth play their games simultaneously. Calculate the probability that Joshua and Ruth will conclude their games in an equal number of tosses of their coins.
   2. Joshua and Ruth play their games simultaneously on 3 occasions. Calculate the probability that, on at least 2 of these occasions, their games will be concluded in an equal number of tosses of their coins. Give your answer to three decimal places.
      (4 marks) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 3}

      \(\boldsymbol { n }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)			\(\frac { 1 } { 8 }\)		\(\frac { 1 } { 16 }\)

      \end{table} \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 4}

      \(\boldsymbol { m }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { M } = \boldsymbol { m } )\)	\(\frac { 1 } { 4 }\)	\(\frac { 3 } { 16 }\)

      \end{table}

5 Aiden takes his car to a garage for its MOT test. The probability that his car will need to have \(X\) tyres replaced is shown in the table.

1. Show that the mean of \(X\) is 1.85 and calculate the variance of \(X\).
2. The charge for the MOT test is \(\pounds c\) and the cost of each new tyre is \(\pounds n\). The total amount that Aiden must pay the garage is \(\pounds T\).
   1. Express \(T\) in terms of \(c , n\) and \(X\).
   2. Hence, using your results from part (a), find expressions for \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).

5 The discrete random variable \(R\) has the following probability distribution.

1. Calculate exact values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
2. 1. By tabulating the probability distribution for \(X = \frac { 1 } { R ^ { 2 } }\), show that \(\mathrm { E } ( X ) = \frac { 25 } { 64 }\).
   2. Hence find the value of the mean of the area of a rectangle which has sides of length \(\frac { 8 } { R }\) and \(\left( R + \frac { 8 } { R } \right)\).
      (3 marks)

3 Morecrest football team always scores at least one goal but never scores more than four goals in each game. The number of goals, \(R\), scored in each game by the team can be modelled by the following probability distribution.

1. Calculate exact values for the mean and variance of \(R\).
2. Next season the team will play 32 games. They expect to win \(90 \%\) of the games in which they score at least three goals, half of the games in which they score exactly two goals and \(20 \%\) of the games in which they score exactly one goal. Find, for next season:
   1. the number of games in which they expect to score at least three goals;
   2. the number of games that they expect to win.