5.02e Discrete uniform distribution

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 \(\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )\)

4 The discrete random variable \(X\) has the distribution \(\mathrm { U } ( n )\).

1. Use the results \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) and \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\) to show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\). It is given that \(\mathrm { E } ( X ) = 13\).
2. Find the value of \(n\).
3. Find \(\mathrm { P } ( X < 7.5 )\). It is given that \(\mathrm { E } ( a X + b ) = 10\) and \(\operatorname { Var } ( a X + b ) = 117\), where \(a\) and \(b\) are positive.
4. Calculate the value of \(a\) and the value of \(b\).

2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).

1. It is given that \(W\) has a Poisson distribution.
   1. Write down the standard deviation of \(W\).
   2. Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
   3. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).

1. The random variable \(W\) has a discrete uniform distribution where

$$\mathrm { P } ( W = w ) = \frac { 1 } { 5 } \quad \text { for } w = 1,2,3,4,5$$

1. Find \(\mathrm { P } ( 2 \leqslant W < 3.5 )\) The discrete random variable \(X = 5 - 2 W\)
2. Find \(\mathrm { E } ( X )\)
3. Find \(\mathrm { P } ( X < W )\) The discrete random variable \(\mathrm { Y } = \frac { 1 } { W }\)
4. Find
   1. the probability distribution of \(Y\)
   2. \(\operatorname { Var } ( Y )\), showing your working.
5. Find \(\operatorname { Var } ( 2 - 3 Y )\)

1. The discrete random variable \(D\) with the following probability distribution represents the score when a 4-sided die is rolled.

1. Write down the name of this distribution. The die is used to play a game and the random variable \(X\) represents the number of points scored. The die is rolled once and if \(D = 2,3\) or 4 then \(X = D\). If \(D = 1\) the die is rolled a second time and \(X = 0\) if \(D = 1\) again, otherwise \(X\) is the sum of the two scores on the die.
2. Show that the probability of scoring 3 points in this game is \(\frac { 5 } { 16 }\)
3. Find the probability of scoring 0 in this game. The table below shows the probability distribution for the remaining values of \(X\).

   \(x\)	0	2	3	4	5
   \(\mathrm { P } ( X = x )\)		\(\frac { 1 } { 4 }\)		\(\frac { 5 } { 16 }\)	\(\frac { 1 } { 16 }\)

4. Find \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\) The discrete random variable \(R\) represents the number of times the die is rolled in the game.
6. Write down the probability distribution of \(R\). The random variable \(Y = 2 R + 0.5\)
7. Show that \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\) The game is played once.
8. Find \(\mathrm { P } ( X > Y )\)

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]

2 The random variable \(T\) has a discrete uniform distribution and takes the values 1, 2, 3, 4 and 5 Find the variance of \(T\) Circle your answer.

5 A spinner has 8 equal areas numbered 1 to 8, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de9f0107-38de-4d0d-8391-4d29b98fa601-06_383_390_319_810} The spinner is spun and lands with one of its edges on the ground. 5

1. Assume that the spinner lands on each number with equal probability. 5
   1. 1. State a distribution that could be used to model the number that the spinner lands on. 5
      2. Use your distribution from part 5
      (a) (i) to find the probability that the spinner lands on a number greater than 5
      [0pt] [1 mark] 5
   2. Clare spins the spinner 1000 times and records the results in the following table.

      Number landed on	1	2	3	4	5	6	7	8
      Frequency	37	64	112	161	308	156	109	53

      5
   3. 1. Explain how the data shows that the model used in part (a) may not be valid.
         5
   4. (ii) Describe how Clare's results could be used to adjust the model.

1 The discrete uniform distribution \(X\) can take values \(1,2,3 , \ldots , 10\) Find \(\mathrm { P } ( X \geq 7 )\) Circle your answer. \(0.3 \quad 0.4 \quad 0.6 \quad 0.7\)

1 The random variable \(T\) follows a discrete uniform distribution and can take values \(1,2,3 , \ldots , 16\) Find the variance of \(T\) Circle your answer.
1.2518 .7521 .2521 .33

3. A string of length 60 cm is cut a random point.

1. Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
2. The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than \(100 \mathrm {~cm} ^ { 2 }\).

In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key 7 s after the star first appears, a simple model of the game assumes that T is a continuous uniform random variable defined over the interval [0, 1].

1. Write down P(T < 0.2). [1]
2. Write down E(T). [1]
3. Use integration to find Var(T). [4]

A group of 20 children each play this game once.

1. Find the probability that no more than 4 children stop the star in less than 0.2 s. [3]

The children are allowed to practise this game so that this continuous uniform model is no longer applicable.

1. Explain how you would expect the mean and variance of T to change. [2]

It is found that a more appropriate model of the game when played by experienced players assumes that T has a probability density function g(t) given by $$g(t) = \begin{cases} 4t, & 0 \leq t \leq 0.5, \\ 4 - 4t, & 0.5 \leq t \leq 1, \\ 0, & otherwise. \end{cases}$$

1. Using this model show that P(T < 0.2) = 0.08. [2]

A group of 75 experienced players each played this game once.

1. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s. [4]

Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable X represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of X and specify it fully. [2]
2. Calculate the mean time of arrival of the bus. [3]
3. Find the cumulative distribution function of X. [4]

Jean will be late for work if the bus arrives after 8.05 a.m.

1. Find the probability that Jean is late for work. [2]

The continuous random variable R is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that E(R) = 3 and Var(R) = \(\frac{4}{3}\), find

1. the value of \(\alpha\) and the value of \(\beta\), [7]
2. P(R < 6.6). [2]

An engineer measures, to the nearest cm, the lengths of metal rods.

1. Suggest a suitable model to represent the difference between the true lengths and the measured lengths. [2]
2. Find the probability that for a randomly chosen rod the measured length will be within 0.2 cm of the true length. [2]

Two rods are chosen at random.

1. Find the probability that for both rods the measured lengths will be within 0.2 cm of their true lengths. [2]

The discrete random variable \(X\) can take any value in the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that P\((X = 1) = 0.125\) and that E\((X) = 4.5\). Beatrice finds that P\((X = 2) =\) P\((X = 3) =\) P\((X = 4) = p\). Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).

1. Calculate the values of \(p\) and \(q\). [7 marks]
2. Give the name for the distribution of \(X\). [1 mark]
3. Calculate the standard deviation of \(X\). [3 marks]

Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.

1. What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution? [1 mark]
2. Making this modelling assumption, find the expectation and the variance of \(X\). [5 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]

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]

A group of 60 children were each asked to choose an integer value between 1 and 9 inclusive. Their choices are summarised in the table below.

Value chosen	1	2	3	4	5	6	7	8	9
Number of children	3	4	5	10	12	13	7	4	2

1. Calculate the mean and standard deviation of the values chosen. [6]

It is suggested that the value chosen could be modelled by a discrete uniform distribution.

1. Write down the mean that this model would predict. [2]

Given also that the standard deviation according to this model would be 2.58,

1. explain why this model is not suitable and suggest why this is the case. [2]

A searchlight is rotating in a horizontal circle. It is assumed that that, at any moment, the centre of its beam is equally likely to be pointing in any direction. The random variable \(X\) represents this direction, expressed as a bearing in the range \(000°\) to \(360°\).

1. Specify a suitable model for the distribution of \(X\). [1 mark]
2. Find the mean and the standard deviation of \(X\). [3 marks]

A child cuts a 30 cm piece of string into two parts, cutting at a random point.

1. Name the distribution of \(L\), the length of the longer part of string, and sketch the probability density function for \(L\). [4 marks]
2. Find the probability that one part of the string is more than twice as long as the other. [3 marks]

The random variable \(X\) has a continuous uniform distribution on the interval \(a \leq X \leq 3a\).

1. Without assuming any standard results, prove that \(\mu\), the mean value of \(X\), is equal to \(2a\) and derive an expression for \(\sigma^2\), the variance of \(X\), in terms of \(a\). [7 marks]
2. Find the probability that \(|X - \mu| < \sigma\) and compare this with the same probability when \(x\) is modelled by a Normal distribution with the same mean and variance. [6 marks]