5.02c Linear coding: effects on mean and variance

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. 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 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. 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. 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 \(\mathrm { P } ( Y = 15 )\)

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

where \(a\) and \(b\) are probabilities.

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 3.9\)
2. find the value of \(a\) and the value of \(b\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
3. Find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 3 \right)\)

1. In a class experiment, each day for 170 days, a child is chosen at random and spins a large cardboard coin 5 times and the number of heads is recorded.
   The results are summarised in the following table.

Marcus believes that a \(\mathrm { B } ( 5,0.5 )\) distribution can be used to model these data and he calculates expected frequencies, to 2 decimal places, as follows

1. Find the value of \(r\) and the value of \(s\)
2. Carry out a suitable test, at the \(5 \%\) level of significance, to determine whether or not the \(\mathrm { B } ( 5,0.5 )\) distribution is a good model for these data.
   You should state clearly your hypotheses, the test statistic and the critical value used. Nima believes that a better model for these data would be \(\mathrm { B } ( 5 , p )\)
3. Find a suitable estimate for \(p\) To test her model, Nima uses this value of \(p\), to calculate expected frequencies as follows

   Number of heads	0	1	2	3	4	5
   Expected frequency	2.07	14.65	41.44	58.63	41.47	11.74

   The test statistic for Nima's test is 1.62 (to 3 significant figures)
4. State,
   1. giving your reasons, the degrees of freedom
   2. the critical value
      that Nima should use for a test at the 5\% significance level.
5. With reference to Marcus' and Nima's test results, comment on
   1. the probability of the coin landing on heads,
   2. the independence of the spins of the coin. Give reasons for your answers.

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

$$\mathrm { G } _ { X } ( t ) = \frac { t ^ { 2 } } { ( 3 - 2 t ) ^ { 2 } }$$

1. Specify the distribution of \(X\) A fair die is rolled repeatedly.
2. Describe an outcome that could be modelled by the random variable \(X\)
3. Use calculus and \(\mathrm { G } _ { X } ( t )\) to find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = \frac { t ^ { 10 } } { \left( 3 - 2 t ^ { 3 } \right) ^ { 2 } }$$
4. Find the exact value of \(\mathrm { P } ( Y = 19 )\)

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

1. Find \(\operatorname { Var } ( X )\)
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\)

1. The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where

$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$

1. Use calculus to find \(\operatorname { Var } ( X )\) Show your working clearly.
2. Find the exact value of \(\mathrm { P } ( X \leqslant 2 )\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\) The random variable \(Y = X _ { 1 } + X _ { 2 } + 1\)
3. By finding the probability generating function of \(Y\), state the name of the distribution of \(Y\)
4. Hence, or otherwise, find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)

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

$$G _ { x } ( t ) = k \left( 3 + t + 2 t ^ { 2 } \right) ^ { 2 }$$

1. Show that \(\mathrm { k } = \frac { 1 } { 36 }\)
2. Find \(\mathrm { P } ( \mathrm { X } = 3 )\)
3. Show that \(\operatorname { Var } ( \mathrm { X } ) = \frac { 29 } { 18 }\)
4. Find the probability generating function of \(2 \mathrm { X } + 1\) \section*{Q uestion 6 continued} \section*{Q uestion 6 continued} \section*{Q uestion 6 continued}

1. Korhan and Louise challenge each other to find an estimator for the mean, \(\mu\), of the continuous random variable \(X\) which has variance \(\sigma ^ { 2 }\) \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) are \(n\) independent observations taken from \(X\) Korhan's estimator is given by

$$K = \frac { 2 } { n ( n + 1 ) } \sum _ { r = 1 } ^ { n } r X _ { r }$$ Louise's estimator is given by $$L = \frac { X _ { 1 } + X _ { 2 } } { 3 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 3 ( n - 2 ) }$$

1. Show that \(K\) and \(L\) are both unbiased estimators of \(\mu\)
2. 1. Find \(\operatorname { Var } ( K )\)
   2. Find \(\operatorname { Var } ( L )\) The winner of the challenge is the person who finds the better estimator.
3. Determine the winner of the challenge for large values of \(n\). Give reasons for your answer.

5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by \(X\).

1. \(\mathrm { E } ( X )\),
2. \(\operatorname { Var } ( X )\).
   1. Write down
   2. Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\).
   3. Use the binomial distribution to find exactly the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\). Show the values of all relevant calculations.

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 \(\operatorname { Var } ( 3 N + 2 )\).

4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } k \left( x ^ { 2 } - 9 \right) , & x = 4,5,6 \\ 0 , & \text { otherwise } \end{array}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Find \(\operatorname { Var } ( 2 X - 3 )\).

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 \(C\).

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 .

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}

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 \(\operatorname { Var } ( 9 A - 2 )\)

1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 5\) Find \(\operatorname { Var } ( 4 X - 3 )\) Circle your answer.
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4 The discrete random variable \(X\) follows a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\). The discrete random variable \(Y\) is defined by \(Y = 2 X\) 4

1. Using the standard results for \(\sum n , \sum n ^ { 2 }\) and \(\operatorname { Var } ( a X + b )\), prove that $$\operatorname { Var } ( Y ) = \frac { n ^ { 2 } - 1 } { 3 }$$ 4
2. A spinning toy can land on one of four values: 2, 4, 6 or 8
   Using a discrete uniform distribution, find the probability that the next value the toy lands on is greater than 2 4
3. State an assumption that is required for the discrete uniform distribution used in part (b) to be valid.

2 The random variable \(X\) has variance \(\operatorname { Var } ( X )\) Which of the following expressions is equal to \(\operatorname { Var } ( a X + b )\), where \(a\) and \(b\) are non-zero constants? Circle your answer.
[0pt] [1 mark] \(a \operatorname { Var } ( X )\) \(a \operatorname { Var } ( X ) + b\) \(a ^ { 2 } \operatorname { Var } ( X )\) \(a ^ { 2 } \operatorname { Var } ( X ) + b\)