5.02c Linear coding: effects on mean and variance

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}

2

1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).

5 A soft drinks company has an automated bottling machine that fills 500 ml bottles with soft drink. The contents of the bottles are measured during a check on the machine. In the check, \(5 \%\) of the bottles contain more than 500 ml and \(2.5 \%\) contain less than 495 ml . It is given that the amount of drink dispensed per bottle is normally distributed.

1. Find the mean and standard deviation of the amount of drink dispensed per bottle, giving your answers to 4 significant figures.
2. It is subsequently found that the measurements of volume made in the checking process are all 3 ml below their true value. Using a corrected distribution, find the probability that a bottle chosen at random contains more than 500 ml of the drink.

It has been found that 60\% of the computer chips produced in a factory are faulty. As part of quality control, 100 samples of 4 chips are selected at random, and each chip is tested. The number of faulty chips in each sample is recorded, with the results given in the following table.

Number of faulty chips	0	1	2	3	4
Number of samples	2	12	27	49	10

The expected values for a binomial distribution with parameters \(n = 4\) and \(p = 0.6\) are given in the following table.

Number of faulty chips	0	1	2	3	4
Expected value	2.56	15.36	34.56	34.56	12.96

Show how the expected value 34.56 corresponding to 2 faulty chips is obtained. [2] Carry out a goodness of fit test at the 5\% significance level, and state what can be deduced from the outcome of the test. [8]

Test scores, \(X\), have mean 54 and variance 144. The scores are scaled using the formula \(Y = a + bX\), where \(a\) and \(b\) are constants and \(b > 0\). The scaled scores, \(Y\), have mean 50 and variance 100. Find the values of \(a\) and \(b\). [4]

Jim records the length, \(l\) mm, of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained. $$n = 81 \quad \sum x = 3711 \quad \sum x^2 = 475181$$

1. Find the mean length of these salmon. [3]
2. Find the variance of the lengths of these salmon. [2]

The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below. \includegraphics{figure_2}

1. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ [1]

Raj says that the box plot is incorrect as Jim has not included outliers. For these data an outlier is defined as a value that is more than \(1.5 \times\) IQR above the upper quartile \quad or \quad \(1.5 \times\) IQR below the lower quartile

1. Show that there are no outliers. [3]

A discrete random variable \(Y\) has probability function $$\mathrm{P}(Y = y) = \begin{cases} k(3 - y) & y = 1, 2 \\ k(y^2 - 8) & y = 3, 4, 5 \\ k & y = 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{30}\) [2]

Find the exact value of

1. P\((1 < Y \leqslant 4)\) [2]
2. E\((Y)\) [2]

The random variable \(X = 15 - 2Y\)

1. Calculate P\((Y \geqslant X)\) [3]
2. Calculate Var\((X)\) [4]

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

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

Find the value of

1. \(E(3X - 2)\), [2]
2. \(\text{Var}(2X + 6)\). [2]

The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} kx, & x = 1, 2, 3, 4, 5, \\ 0, & \text{otherwise.} \end{cases}$$

1. Show that \(k = \frac{1}{15}\). [3]

Find the value of

1. E\((2X + 3)\), [5]
2. Var\((2X - 4)\). [6]

The variable \(X\) represents the marks out of 150 scored by a group of students in an examination. The following ten values of \(X\) were obtained: 60, 66, 76, 80, 94, 106, 110, 116, 124, 140.

1. Write down the median, \(M\), of the ten marks. [1 mark]
2. Using the coding \(y = \frac{x - M}{2}\), and showing all your working clearly, find the mean and the standard deviation of the marks. [6 marks]
3. Find E\((3X - 5)\). [3 marks]

The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} cx^2 & x = -3, -2, -1, 1, 2, 3 \\ 0 & \text{otherwise.} \end{cases}$$

1. Show that \(c = \frac{1}{28}\). [3 marks]
2. Calculate
   1. \(E(X)\),
   2. \(E(X^2)\).
   [3 marks]
3. Calculate
   1. \(\text{Var}(X)\),
   2. \(\text{Var}(10 - 2X)\).
   [3 marks]

The random variable \(X\), which can take any value from \(\{1, 2, \ldots, n\}\), is modelled by the discrete uniform distribution with mean 10.

1. Show that \(n = 19\) and find the variance of \(X\). [4 marks]
2. Find \(\text{P}(3 < X \leq 6)\). [2 marks]

The random variable \(Y\) is defined by \(Y = 3(X - 10)\).

1. State the mean and the variance of \(Y\). [3 marks]

The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\text{f}(x) = \begin{cases} k(x + 1) & x = 1, 2, \ldots, 19, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{1}{209}\). [3 marks]
2. Find \(\text{P}(3 < X \leq 6)\) using this model. [3 marks]

Using the coding \(y = \frac{x-90}{5}\), and showing each step in your working clearly, calculate the mean and the standard deviation of the 20 observations of a variable \(X\) given by the following table:

\(x\)	75	80	85	90	95	100	105	110
Frequency	1	2	3	6	4	2	1	1

[8 marks]

A pack of 52 cards contains 4 cards bearing each of the integers from 1 to 13. A card is selected at random. The random variable \(X\) represents the number on the card.

1. Find \(P(X \leq 5)\). [1 mark]
2. Name the distribution of \(X\) and find the expectation and variance of \(X\). [4 marks]

A hand of 12 cards consists of three 2s, four 3s, two 4s, two 5s and one 6. The random variable \(Y\) represents the number on a card chosen at random from this hand.

1. Draw up a table to show the probability distribution of \(Y\). [3 marks]
2. Calculate \(\text{Var}(3Y - 2)\). [6 marks]

The random variable \(X\) has the discrete uniform distribution and takes the values \(\{1, \ldots, n\}\). The standard deviation of of \(X\) is \(2\sqrt{6}\). Find

1. the mean of \(X\), [3 marks]
2. P\((3 \leq X < \frac{2}{5}n)\). [3 marks]

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:

\(x\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)
P\((X = x)\)	\(p\)	\(0.1\)	\(q\)	\(q\)	\(0.3\)	\(0.2\)

It is known that E\((X) = 6.7\). Find

1. the values of \(p\) and \(q\), [7 marks]
2. the value of \(a\) for which E\((2X + a) = 0\), [3 marks]
3. Var\((X)\). [3 marks]

A regular tetrahedron has its faces numbered 1, 2, 3 and 4. It is weighted so that when it is thrown, the probability of each face being in contact with the table is inversely proportional to the number on that face. This number is represented by the random variable \(X\).

1. Show that \(P(X = 1) = \frac{12}{25}\) and find the probabilities of the other values of \(X\). [5 marks]
2. Calculate the mean and the variance of \(X\). [5 marks]

A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.

\(x\)	0	1	2	3
P\((X = x)\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(\frac{1}{8}\)	\(\frac{1}{8}\)

1. Calculate E\((X)\) and Var\((X)\). [5]

The die is thrown 10 times.

1. Find the probability that there are not more than 4 throws on which the score is 1. [2]
2. Find the probability that there are exactly 4 throws on which the score is 2. [3]

When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values 2, 4, 6 and 8. The spinner is biased so that P(\(X = x\)) = \(kx\), where \(k\) is a constant.

1. Show that P(\(X = 6\)) = \(\frac{3}{10}\). [2]
2. Find E(\(X\)) and Var(\(X\)). [5]

A random variable \(X\) has the distribution B\((5, \frac{1}{4})\).

1. Find
   1. E(\(X\)), [1]
   2. P(\(X = 2\)). [2]
2. Two values of \(X\) are chosen at random. Find the probability that their sum is less than 2. [4]
3. 10 values of \(X\) are chosen at random. Use an appropriate formula to find the probability that exactly 3 of these values are 2s. [3]

20% of packets of a certain kind of cereal contain a free gift. Jane buys one packet a week for 8 weeks. The number of free gifts that Jane receives is denoted by \(X\). Assuming that Jane's 8 packets can be regarded as a random sample, find

1. P(\(X = 3\)), [3]
2. P(\(X \geqslant 3\)), [2]
3. E(\(X\)). [2]

Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is 0.12.

1. Find the smallest value of \(n\) such that the probability of at least one success in \(n\) trials is more than 0.95. [3]
2. Find the probability that the 3rd success occurs on the 7th trial. [5]

At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m\) kg, and for each person the value of \((m - 5)\) was recorded. The mean and standard deviation of \((m - 5)\) were found to be 0.74 and 0.13 respectively.

1. Write down the mean and standard deviation of \(m\). [2]

The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.

1. Calculate the mean of all 25 estimates. [2]
2. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg, comment on this claim. [2]

The incomes of a sample of 918 households on an island are given in the table below.

1. Draw a histogram to illustrate the data. [5]
2. Calculate an estimate of the mean income. [3]
3. Calculate an estimate of the standard deviation of the incomes. [4]
4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation. [4]
5. The incomes were converted into another currency using the formula \(y = 1.15x\). Calculate estimates of the mean and variance of the incomes in the new currency. [3]