5.02b Expectation and variance: discrete random variables

1. The probability distribution for the prize money, \(\pounds X\) per ticket, in a local fundraising lottery is shown below.

1. Calculate the value of \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. 1. What is the minimum lottery ticket price that the organiser should set in order to make a profit in the long run?
   2. Suggest why, in practice, people would be prepared to pay more than this minimum price.
      

6. Penelope makes 8 cakes per week. Each cake costs \(\pounds 20\) to make and sells for \(\pounds 60\). She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is \(0 \cdot 3\). She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is \(\pounds 206\). Construct a probability distribution for the weekly profit.
Additional page, if required. number Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

1. A fair six-sided black die has faces numbered \(1,2,2,3,3\) and 4

The random variable \(B\) represents the score when the black die is rolled.

1. Write down the value of \(\mathrm { E } ( B )\) A white die has 6 faces numbered \(1,1,2,4,5\) and \(c\) where \(c > 5\) The discrete random variable \(W\) represents the score when the white die is rolled and has probability distribution given by

   \(w\)	1	2	4	5	\(c\)
   \(\mathrm { P } ( W = w )\)	\(a + b\)	\(a\)	0.3	\(a\)	\(b\)

   Greg and Nilaya play a game with these dice.
   Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
   The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
2. Find the value of \(a\) and the value of \(b\) Show your working clearly. The random variable \(X = 2 W - 5\) Given that \(\mathrm { E } ( X ) = 2.6\)
3. find the exact value of \(\operatorname { Var } ( X )\)

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

where \(q\) and \(r\) are probabilities.

1. Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\) Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
3. find the value of \(q\) and the value of \(r\)
4. Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)

1. In an experiment, James flips a coin 3 times and records the number of heads. He carries out the experiment 100 times with his left hand and 100 times with his right hand.

1. Test, at the \(5 \%\) level of significance, whether or not there is an association between the hand he flips the coin with and the number of heads. You should state your hypotheses, the degrees of freedom and the critical value used for this test.
2. Assuming the coin is unbiased, write down the distribution of the number of heads in 3 flips.
3. Carry out a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether or not the distribution you wrote down in part (b) is a suitable model for the number of heads obtained in the 200 trials of James' experiment. You should state your hypotheses, the degrees of freedom and the critical value used for this test.

1. The probability distribution of the discrete random variable \(X\) is

$$P ( X = x ) = \begin{cases} \frac { k } { x } & \text { for } x = 1,2 \text { and } 3 \\ \frac { m } { 2 x } & \text { for } x = 6 \text { and } 9 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(m\) are positive constants.
Given that \(\mathrm { E } ( X ) = 3.8\), find \(\operatorname { Var } ( X )\)

1. Flobee sells tomato seeds in packets, each containing 40 seeds. Flobee advertises that only 4\% of its tomato seeds do not germinate.

Amodita is investigating the germination of Flobee's tomato seeds. She plants 125 packets of Flobee's tomato seeds and records the number of seeds that do not germinate in each packet. Amodita wants to test whether the binomial distribution \(\mathrm { B } ( 40,0.04 )\) is a suitable model for these data. The table below shows the expected frequencies, to 2 decimal places, using this model.

1. Calculate the value of \(r\) and the value of \(s\)
2. Stating your hypotheses clearly, carry out the test at the \(5 \%\) level of significance. You should state the number of degrees of freedom, critical value and conclusion clearly. Amodita believes that Flobee should use a more realistic value for the percentage of their tomato seeds that do not germinate.
   She decides to test the data using a new model \(\mathrm { B } ( 40 , p )\)
3. Showing your working, suggest a more realistic value for \(p\)

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

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 8.79\)
2. find \(\mathrm { E } \left( X ^ { 2 } \right)\) The discrete random variable \(Y\) has probability distribution

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

   where \(a\), \(b\) and \(c\) are constants.
   For the random variable \(Y\) $$\mathrm { P } ( Y \leqslant 0 ) = 0.75 \quad \text { and } \quad \mathrm { E } \left( Y ^ { 2 } + 3 \right) = 5$$
3. Find the value of \(a\), the value of \(b\) and the value of \(c\) The random variable \(W = Y - X\) where \(Y\) and \(X\) are independent.
   The random variable \(T = 3 W - 8\)
4. Calculate \(\mathrm { P } ( W > T )\)

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

1. Find in terms of \(q\)
   1. \(\mathrm { E } ( X )\)
   2. \(\mathrm { E } \left( X ^ { 2 } \right)\) Given that \(\operatorname { Var } ( X ) = 3.66\)
2. show that \(q = 0.3\) In a game, the score is given by the discrete random variable \(X\) Given that games are independent,
3. calculate the probability that after the 4th game has been played, the total score is exactly 20 A round consists of 4 games plus 2 bonus games. The bonus games are only played if after the 4th game has been played the total score is exactly 20 A prize of \(\pounds 10\) is awarded if 6 games are played in a round and the total score for the round is at least 27 Bobby plays 3 rounds.
4. Find the probability that Bobby wins at least \(\pounds 10\)

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

where \(r\) and \(k\) are positive constants.
The standard deviation of \(X\) equals the mean of \(X\) Find the exact value of \(r\)

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

where \(p\) and \(r\) are probabilities.
Given that \(\mathrm { E } ( X ) = 1.95\) find the exact value of \(\mathrm { E } ( \sqrt { X + 1 } )\) giving your answer in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.
(6)

1. Robin shoots 8 arrows at a target each day for 100 days.

The number of times he hits the target each day is summarised in the table below. Misha believes that these data can be modelled by a binomial distribution.

1. State, in context, two assumptions that are implied by the use of this model.
2. Find an estimate for the proportion of arrows Robin shoots that hit the target. Misha calculates expected frequencies, to 2 decimal places, as follows.

   Number of hits	0	1	2	3	4	5	6	7	8
   Expected frequency	2.81	12.67	\(r\)	28.05	19.73	\(s\)	2.50	0.40	0.03

3. Find the value of \(r\) and the value of \(s\) Misha correctly used a suitable test to assess her belief.
4. 1. Explain why she used a test with 3 degrees of freedom.
   2. Complete the test using a \(5 \%\) level of significance. You should clearly state your hypotheses, test statistic, critical value and conclusion.

1. The discrete random variable \(X\) has probability distribution given by

The random variable \(Y = 2 - 5 X\) Given that \(\mathrm { E } ( \mathrm { Y } ) = - 4\) and \(\mathrm { P } ( \mathrm { Y } \geqslant - 3 ) = 0.45\)

1. find the probability distribution of X . Given also that \(\mathrm { E } \left( \mathrm { Y } ^ { 2 } \right) = 75\)
2. find the exact value of \(\operatorname { Var } ( \mathrm { X } )\)
3. Find \(\mathrm { P } ( \mathrm { Y } > \mathrm { X } )\) \section*{Q uestion 2 continued}

1. Two car hire companies hire cars independently of each other.

Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.

1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
   In a random sample of 600 new cars produced at the factory,
4. 1. find the mean of the number of cars with a defect,
   2. find the variance of the number of cars with a defect.
5. 1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
   2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}

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

$$\mathrm { G } _ { X } ( t ) = k \ln \left( \frac { 2 } { 2 - t } \right)$$ where \(k\) is a constant.

1. Find the exact value of \(k\)
2. Find the exact value of \(\operatorname { Var } ( X )\)
3. Find \(\mathrm { P } ( X = 3 )\)

1. A spinner can land on red or blue. When the spinner is spun, there is a probability of \(\frac { 1 } { 3 }\) that it lands on blue. The spinner is spun repeatedly.

The random variable \(B\) represents the number of the spin when the spinner first lands on blue.

1. Find (i) \(\mathrm { P } ( B = 4 )\) (ii) \(\mathrm { P } ( B \leqslant 5 )\)
2. Find \(\mathrm { E } \left( B ^ { 2 } \right)\) Steve invites Tamara to play a game with this spinner.
   Tamara must choose a colour, either red or blue.
   Steve will spin the spinner repeatedly until the spinner first lands on the colour Tamara has chosen. The random variable \(X\) represents the number of the spin when this occurs. If Tamara chooses red, her score is \(\mathrm { e } ^ { X }\) If Tamara chooses blue, her score is \(X ^ { 2 }\)
3. State, giving your reasons and showing any calculations you have made, which colour you would recommend that Tamara chooses.

1. The discrete random variables \(W , X\) and \(Y\) are distributed as follows

$$W \sim \mathrm {~B} ( 10,0.4 ) \quad X \sim \operatorname { Po } ( 4 ) \quad Y \sim \operatorname { Po } ( 3 )$$

1. Explain whether or not \(\mathrm { Po } ( 4 )\) would be a good approximation to \(\mathrm { B } ( 10,0.4 )\)
2. State the assumption required for \(X + Y\) to be distributed as \(\operatorname { Po } ( 7 )\) Given the assumption in part (b) holds,
3. find \(\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )\)

1. Suzanne and Jon are playing a game.

They put 4 red counters and 1 blue counter in a bag.
Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.

1. Find the probability that Suzanne selects the blue counter on her 4th selection.
2. Find the probability that the blue counter is first selected on or after Jon's third selection.
3. Find the mean and standard deviation of the number of selections made until the blue counter is selected.
4. Find the probability that Suzanne wins the game.

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

1. Find \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) is defined in terms of the discrete random variable \(X\) When \(X\) is negative, \(Y = X ^ { 2 }\) When \(X\) is positive, \(Y = 3 X - 2\)
2. Find \(\mathrm { P } ( Y < 9 )\)
3. Find \(\mathrm { E } ( X Y )\)

1. A discrete random variable \(X\) has probability generating function given by

$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 64 } \left( a + b t ^ { 2 } \right) ^ { 2 }$$ where \(a\) and \(b\) are positive constants.

1. Write down the value of \(\mathrm { P } ( X = 3 )\) Given that \(\mathrm { P } ( X = 4 ) = \frac { 25 } { 64 }\)
2. 1. find \(\mathrm { P } ( X = 2 )\)
   2. find \(\mathrm { E } ( X )\) The random variable \(Y = 3 X + 2\)
3. Find the probability generating function of \(Y\)

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.

1. Asha, Davinda and Jerry each have a bag containing a large number of counters, some of which are white and the rest are red.
   Each person draws counters from their bag one at a time, notes the colour of the counter and returns it to their bag.

The probability of Asha getting a red counter on any one draw is 0.07

1. Find the probability that Asha will draw at least 3 white counters before a red counter is drawn.
2. Find the probability that Asha gets a red counter for the second time on her 9th draw. The probability of Davinda getting a red counter on any one draw is \(p\). Davinda draws counters until she gets \(n\) red counters. The random variable \(D\) is the number of counters Davinda draws. Given that the mean and the standard deviation of \(D\) are 4400 and 660 respectively,
3. find the value of \(p\). Jerry believes that his bag contains a smaller proportion of red counters than Asha's bag. To test his belief, Jerry draws counters from his bag until he gets a red counter. Jerry defines the random variable \(J\) to be the number of counters drawn up to and including the first red counter.
4. Stating your hypotheses clearly and using a \(10 \%\) level of significance, find the critical region for this test. Jerry gets a red counter for the first time on his 34th draw.
5. Giving a reason for your answer, state whether or not there is evidence that Jerry's bag contains a smaller proportion of red counters than Asha's bag. Given that the probability of Jerry getting a red counter on any one draw is 0.011
6. show that the power of the test is 0.702 to 3 significant figures.

1. The probability generating function of the random variable \(X\) is

$$\mathrm { G } _ { X } ( t ) = k ( 1 + 2 t ) ^ { 5 }$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 243 }\)
2. Find \(\mathrm { P } ( X = 2 )\)
3. Find the probability generating function of \(W = 2 X + 3\) The probability generating function of the random variable \(Y\) is $$\mathrm { G } _ { Y } ( t ) = \frac { t ( 1 + 2 t ) ^ { 2 } } { 9 }$$ Given that \(X\) and \(Y\) are independent,
4. find the probability generating function of \(U = X + Y\) in its simplest form.
5. Use calculus to find the value of \(\operatorname { Var } ( U )\)

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

where \(b\) is a constant and \(b > 5\)

1. Find \(\mathrm { E } ( X )\) in terms of \(b\) Given that \(\operatorname { Var } ( X ) = 34.26\)
2. find the value of \(b\)
3. Find \(\mathrm { P } \left( X ^ { 2 } < 2 - 3 X \right)\)

1. The discrete random variable \(V\) has probability distribution

1. Show that the probability generating function of \(V\) is $$\mathrm { G } _ { V } ( t ) = t ^ { 2 } \left( \frac { 2 } { 5 } t + \frac { 3 } { 5 } \right) ^ { 2 }$$ The discrete random variable \(W\) has probability generating function $$\mathrm { G } _ { W } ( t ) = t \left( \frac { 2 } { 5 } t + \frac { 3 } { 5 } \right) ^ { 5 }$$
2. Use calculus to find
   1. \(\mathrm { E } ( W )\)
   2. \(\operatorname { Var } ( W )\) Given that \(V\) and \(W\) are independent,
3. find the probability generating function of \(X = V + W\) in its simplest form. The discrete random variable \(Y = 2 X + 3\)
4. Find the probability generating function of \(Y\)
5. Find \(\mathrm { P } ( Y = 15 )\)