5.02a Discrete probability distributions: general

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. 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. 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 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. 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 \(V\) 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. 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. A rectangle is to have an area of \(40 \mathrm {~cm} ^ { 2 }\)

The length of the rectangle, \(L \mathrm {~cm}\), follows a continuous uniform distribution over the interval [4, 10] Find the expected value of the perimeter of the rectangle.
Use algebraic integration, rather than your calculator, to evaluate any definite integrals.

3 In this question you must show detailed reasoning. A discrete random variable \(V\) has the following probability distribution, where \(p\) and \(q\) are constants. It is given that \(\mathrm { E } ( V ) = \operatorname { Var } ( V )\). Determine the value of \(p\) and the value of \(q\).

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

2 In a fairground game a competitor scores \(0,1,2\) or 3 with probabilities given in the following table, where \(a\) and \(b\) are constants. The competitor's expected score is 0.9 .

1. Show that \(b = 0.15\).
2. Find the variance of the score.
3. The competitor has to pay \(\pounds 2.50\) to take part, and wins a prize of \(\pounds 2 X\), where \(X\) is the score achieved. Find the expectation of the competitor's loss.

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 \(\mathrm { P } ( X > Y )\)

4. Three bags A, B and \(\mathbf { C }\) each contain coloured balls. Bag A contains 4 red balls and 2 yellow balls only.
Bag B contains 4 red balls and 1 yellow ball only.
Bag \(\mathbf { C }\) contains 6 red balls only. In a game
Mike takes a ball at random from bag \(\mathbf { A }\), records the colour and places it in bag \(\mathbf { C }\). He then takes a ball at random from bag \(\mathbf { B }\), records the colour and places it in bag \(\mathbf { C }\). Finally, Mike takes a ball at random from bag \(\mathbf { C }\) and records the colour.

1. Complete the tree diagram on the page opposite, to illustrate the game by adding the remaining branches and all probabilities.
2. Show that the probability that Mike records a yellow ball exactly twice is \(\frac { 1 } { 10 }\) Given that Mike records exactly 2 yellow balls,
3. find the probability that the ball drawn from bag \(\mathbf { A }\) is red. Mike plays this game a large number of times, each time starting with the bags containing balls as described above. The random variable \(X\) represents the number of yellow balls recorded in a single game.
4. Find the probability distribution of \(X\)
5. Find \(\mathrm { E } ( X )\) Bag B
   Bag C \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Bag A} \includegraphics[alt={},max width=\textwidth]{29ac0c0b-f963-40a1-beba-7146bbb2d021-13_739_1580_411_182}
   \end{figure}

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

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

1. Write down the value of \(\mathrm { E } ( Y )\) The cumulative distribution function of \(Y\) is \(\mathrm { F } ( y )\) Given that \(F ( 0 ) = \frac { 19 } { 30 }\)
2. show that the value of \(u\) is \(\frac { 4 } { 15 }\) Given also that \(\operatorname { Var } ( Y ) = 37\)
3. find the value of \(q\) and the value of \(r\) The coordinates of a point \(P\) are \(( 12 , Y )\) The random variable \(D\) represents the length of \(O P\)
4. Find the probability distribution of \(D\)

7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.

1. Given that \(\mathrm { E } ( X ) = - 0.2\), find the value of \(\alpha\) and the value of \(\beta\).
2. Write down \(\mathrm { F } ( 0.8 )\).
   1. Evaluate \(\operatorname { Var } ( X )\).

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 \(\operatorname { Var } ( 2 X - 3 )\).

5. The random variable \(X\) represents the number on the uppermost face when a fair die is thrown.

1. Write down the name of the probability distribution of \(X\).
2. Calculate the mean and the variance of \(X\). Three fair dice are thrown and the numbers on the uppermost faces are recorded.
3. Find the probability that all three numbers are 6 .
4. Write down all the different ways of scoring a total of 16 when the three numbers are added together.
5. Find the probability of scoring a total of 16 .

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 \(C\).