5.02e Discrete uniform distribution

7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
   In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
2. Calculate the probability of a Type I error.
   Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
3. 1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
   2. Justify your approximating distribution in this context.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 The random variable \(X\) has a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\) The mean of \(X\) is 8 3

1. Show that \(n = 15\) [0pt] [2 marks]
   LL
   3
2. \(\quad\) Find \(\mathrm { P } ( X > 4 )\) 3
3. Find the variance of \(X\), giving your answer in exact form.

3 The discrete random variable \(W\) has the distribution \(\mathrm { U } ( 11 )\). The independent discrete random variable \(V\) has the distribution \(\mathrm { U } ( 5 )\).

1. It is given that, for constants \(m\) and \(n\), with \(m > 0\), \(\mathrm { E } ( \mathrm { mW } + \mathrm { nV } ) = 0\) and \(\operatorname { Var } ( \mathrm { mW } + \mathrm { nV } ) = 1\). Determine the exact values of \(m\) and \(n\). The random variable \(T\) is the mean of three independent observations of \(W\).
2. Explain whether the Central Limit Theorem can be used to say that the distribution of \(T\) is approximately normal.

3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$

1. Write down the name given to this distribution. Find
2. \(\mathrm { P } ( X = 4 )\)
3. \(\mathrm { F } ( 3 )\)
4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
5. Write down \(\mathrm { E } ( X )\)
6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
8. find \(\operatorname { Var } ( a X - 3 )\)

6. The random variable \(A\) represents the score when a spinner is spun. The probability distribution for \(A\) is given in the following table.

1. Show that \(\mathrm { E } ( A ) = 3.5\)
2. Find \(\operatorname { Var } ( A )\) The random variable \(B\) represents the score on a 4 -sided die. The probability distribution for \(B\) is given in the following table where \(k\) is a positive integer.

   \(b\)	1	3	4	\(k\)
   \(\mathrm { P } ( B = b )\)	0.25	0.25	0.25	0.25

3. Write down the name of the probability distribution of \(B\).
4. Given that \(\mathrm { E } ( B ) = \mathrm { E } ( A )\) state, giving a reason, the value of \(k\). The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Sam and Tim are playing a game with the spinner and the die. They each spin the spinner once to obtain their value of \(A\) and each roll the die once to obtain their value of \(B\).
   Their value of \(A\) is taken as their value of \(\mu\) and their value of \(B\) is taken as their value of \(\sigma\). The person with the larger value of \(\mathrm { P } ( X > 3.5 )\) is the winner.
5. Given that Sam obtained values of \(a = 4\) and \(b = 3\) and Tim obtained \(b = 4\) find, giving a reason, the probability that Tim wins.
6. Find the largest value of \(\mathrm { P } ( X > 3.5 )\) achievable in this game.
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3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$

1. Write down the name given to this distribution. Find
2. \(\mathrm { P } ( X = 4 )\)
3. \(\mathrm { F } ( 3 )\)
4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
5. Write down \(\mathrm { E } ( X )\)
6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
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6. A discrete random variable is such that each of its values is assumed to be equally likely.

1. Write down the name of the distribution that could be used to model this random variable.
2. Give an example of such a distribution.
3. Comment on the assumption that each value is equally likely.
4. Suggest how you might refine the model in part (a).

5. (a) Write down two reasons for using statistical models.
(b) Give an example of a random variable that could be modelled by

1. a normal distribution,
2. a discrete uniform distribution.

6. A fair blue die has faces numbered \(1,1,3,3,5\) and 5 . The random variable \(B\) represents the score when the blue die is rolled.

1. Write down the probability distribution for \(B\).
2. State the name of this probability distribution.
3. Write down the value of \(\mathrm { E } ( B )\). A second die is red and the random variable \(R\) represents the score when the red die is rolled. The probability distribution of \(R\) is

   \(r\)	2	4	6
   \(\mathrm { P } ( R = r )\)	\(\frac { 2 } { 3 }\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 6 }\)

4. Find \(\mathrm { E } ( R )\).
5. Find \(\operatorname { Var } ( R )\). Tom invites Avisha to play a game with these dice.
   Tom spins a fair coin with one side labelled 2 and the other side labelled 5 . When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is greater than the number showing on the coin, Avisha wins, otherwise Tom wins. Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.
6. Find the probability that Avisha wins the game, stating clearly which die she should use in each case.

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

$$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } , \quad x = 1,2,3,4,5$$

1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 2\). Find
2. \(\mathrm { E } ( 3 X - 2 )\),
3. \(\operatorname { Var } ( 4 - 3 X )\).

5. The random variable \(X\) has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that \(\mathrm { E } ( X ) = 5\),

1. show that \(n = 9\). Find
2. \(\mathrm { P } ( X < 7 )\),
3. \(\operatorname { Var } ( X )\).

1. A point is to be randomly plotted on the \(x\)-axis, where the units are measured in cm .

The random variable \(R\) represents the \(x\) coordinate of the point on the \(x\)-axis and \(R\) is uniformly distributed over the interval [-5,19] A negative value indicates that the point is to the left of the origin and a positive value indicates that the point is to the right of the origin.

1. Find the exact probability that the point is plotted to the right of the origin.
2. Find the exact probability that the point is plotted more than 3.5 cm away from the origin.
3. Sketch the cumulative distribution function of \(R\) Three independent points with \(x\) coordinates \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) are plotted on the \(x\)-axis.
4. Find the exact probability that
   1. all three points are more than 10 cm from the origin
   2. the point furthest from the origin is more than 10 cm from the origin.

1. Akia selects at random a value from the continuous random variable \(W\), which is uniformly distributed over the interval \([ a , b ]\)

The probability that Akia selects a value greater than 17 is \(\frac { 1 } { 5 }\) The probability that Akia selects a value less than \(k\) is \(\frac { 53 } { 60 }\)

1. Find the probability that Akia selects a value between 17 and \(k\) It is known that \(\operatorname { Var } ( W ) = 75\)
2. 1. Find the value of \(a\) and the value of \(b\)
   2. Find the value of \(k\)
3. Find \(\mathrm { P } ( - 5 < W < 5 )\)
4. Find \(\mathrm { E } \left( W ^ { 2 } \right)\)

3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).

1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( \mathrm { X } )\),
4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).

2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2 \\ \frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4 \\ 1 , & x > 4 \end{cases}$$

1. Find \(\mathrm { P } ( X < 0 )\).
2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
3. Write down the name of the distribution of \(X\).
4. Find the mean and the variance of \(X\).
5. Write down the value of \(\mathrm { P } ( X = 1 )\).

7. 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 \(T\) 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 \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
2. Write down E(T).
3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5 \\ 4 - 4 t , & 0.5 \leq t \leq 1 , \\ 0 , & \text { otherwise } . \end{cases}$$
6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
   (4) END

2. The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).

1. Write down the probability density function \(\mathrm { f } ( x )\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( X )\),
4. the cumulative distribution function of \(X\), for all \(x\),
5. \(\mathrm { P } ( 2.3 < X < 3.4 )\).

3. The random variable \(X\) has a continuous uniform distribution on \([ a , b ]\) where \(a\) and \(b\) are positive numbers. Given that \(\mathrm { E } ( X ) = 23\) and \(\operatorname { Var } ( X ) = 75\)

1. find the value of \(a\) and the value of \(b\). Given that \(\mathrm { P } ( X > c ) = 0.32\)
2. find \(\mathrm { P } ( 23 < X < c )\).

1. In a village shop the customers must join a queue to pay. The number of customers joining the queue in a 10 minute interval is modelled by a Poisson distribution with mean 3

Find the probability that

1. exactly 4 customers join the queue in the next 10 minutes,
2. more than 10 customers join the queue in the next 20 minutes. When a customer reaches the front of the queue the customer pays the assistant. The time each customer takes paying the assistant, \(T\) minutes, has a continuous uniform distribution over the interval \([ 0,5 ]\). The random variable \(T\) is independent of the number of people joining the queue.
3. Find \(\mathrm { P } ( T > 3.5 )\) In a random sample of 5 customers, the random variable \(C\) represents the number of customers who took more than 3.5 minutes paying the assistant.
4. Find \(\mathrm { P } ( C \geqslant 3 )\) Bethan has just reached the front of the queue and starts paying the assistant.
5. Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.

7. A piece of string \(A B\) has length 9 cm . The string is cut at random at a point \(P\) and the random variable \(X\) represents the length of the piece of string \(A P\).

1. Write down the distribution of \(X\).
2. Find the probability that the length of the piece of string \(A P\) is more than 6 cm . The two pieces of string \(A P\) and \(P B\) are used to form two sides of a rectangle. The random variable \(R\) represents the area of the rectangle.
3. Show that \(R = a X ^ { 2 } + b X\) and state the values of the constants \(a\) and \(b\).
4. Find \(\mathrm { E } ( R )\).
5. Find the probability that \(R\) is more than twice the area of a square whose side has the length of the piece of string \(A P\).

4. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) Given that \(\mathrm { E } ( X ) = 3.5\) and \(\mathrm { P } ( X > 5 ) = \frac { 2 } { 5 }\)

1. find the value of \(\alpha\) and the value of \(\beta\) Given that \(\mathrm { P } ( X < c ) = \frac { 2 } { 3 }\)
2. 1. find the value of \(c\)
   2. find \(\mathrm { P } ( c < X < 9 )\) A rectangle has a perimeter of 200 cm . The length, \(S \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 30 cm and 80 cm .
3. Find the probability that the length of the shorter side of the rectangle is less than 45 cm .

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 \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).