Discrete Probability Distributions

2 A flower shop has 5 yellow roses, 3 red roses and 2 white roses. Martin chooses 3 roses at random. Draw up the probability distribution table for the number of white roses Martin chooses.

3 John plays a game with two unbiased coins. John tosses the coins. If he gets two heads he wins \(\pounds 1\). If he gets two tails he wins 20 p. If he gets one head and one tail he wins nothing. Let \(X\) be the random variable for the amount of money, in pence, John wins per game.

1. Construct a probability distribution table for \(X\).
2. Calculate \(\mathrm { E } ( X )\).
3. John pays \(s\) pence to play the game. State the values of \(s\) for which John should expect to make a loss.

6. Penelope makes 8 cakes per week. Each cake costs \(\pounds 20\) to make and sells for \(\pounds 60\). She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is \(0 \cdot 3\). She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is \(\pounds 206\). Construct a probability distribution for the weekly profit.
Additional page, if required. number Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

9 The random variable \(X\) has probability distribution given by $$\mathrm { P } ( X = x ) = a + b x \quad \text { for } x = 1,2 \text { and } 3 ,$$ where \(a\) and \(b\) are constants.

1. Show that \(3 a + 6 b = 1\).
2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 3 }\), find \(a\) and \(b\).

1 A competitor in a throwing event has three attempts to throw a ball as far as possible. The random variable \(X\) denotes the number of throws that exceed 30 metres. The probability distribution table for \(X\) is shown below.

1. Given that \(\mathrm { E } ( X ) = 1.1\), find the value of \(p\) and the value of \(r\).
2. Find the numerical value of \(\operatorname { Var } ( X )\).

1. Tisam is playing a game.

She uses a ball, a cup and a spinner.
The random variable \(X\) represents the number the spinner lands on when it is spun. The probability distribution of \(X\) is given in the following table where \(a , b , c\) and \(d\) are probabilities.
To play the game

• the spinner is spun to obtain a value of \(x\)
• Tisam then stands \(x \mathrm {~cm}\) from the cup and tries to throw the ball into the cup

The event \(S\) represents the event that Tisam successfully throws the ball into the cup.
To model this game Tisam assumes that

• \(\mathrm { P } ( S \mid \{ X = x \} ) = \frac { k } { x }\) where \(k\) is a constant
• \(\mathrm { P } ( S \cap \{ X = x \} )\) should be the same whatever value of \(x\) is obtained from the spinner

Using Tisam's model,

1. show that \(c = \frac { 8 } { 5 } b\)
2. find the probability distribution of \(X\) Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm .
   He successfully gets the ball in the cup \(30 \%\) of the time.
3. State, giving a reason, why Tisam's model of this game is not suitable to describe Nav playing the game for all values of \(X\)

2 The random variable \(X\) takes the values \(- 2,0\) and 4 only. It is given that \(\mathrm { P } ( X = - 2 ) = 2 p , \mathrm { P } ( X = 0 ) = p\) and \(\mathrm { P } ( X = 4 ) = 3 p\).

1. Find \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 Part of the probability distribution of a variable, \(X\), is given in the table.

1. Find \(\mathrm { P } ( X = 0 )\).
2. Find \(\mathrm { E } ( X )\).

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

where

• \(\quad a , b\) and \(c\) are distinct integers \(( a < b < c )\)
• all the probabilities are greater than zero
  1. Find
     1. the value of a
     2. the value of \(b\)
     3. the value of \(c\)

Show your working clearly. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)

Find \(\mathrm { P } \left( X _ { 1 } = X _ { 2 } \right)\) \section*{Question 6 continued.} \section*{Question 6 continued.}

5 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4$$

1. Show that \(k = 0.05\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 The random variable \(X\) takes the values \(- 2,1,2,3\). It is given that \(\mathrm { P } ( X = x ) = k x ^ { 2 }\), where \(k\) is a constant.

1. Draw up the probability distribution table for \(X\), giving the probabilities as numerical fractions.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

8. The discrete random variable \(R\) takes even integer values from 2 to \(2 n\) inclusive.
The probability distribution of \(R\) is given by $$\mathrm { P } ( R = r ) = \frac { r } { k } \quad r = 2,4,6 , \ldots , 2 n$$ where \(k\) is a constant.

1. Show that \(k = n ( n + 1 )\) When \(n = 20\)
2. find the exact value of \(\mathrm { P } ( 16 \leqslant R < 26 )\) When \(n = 20\), a random value \(g\) of \(R\) is taken and the quadratic equation in \(x\) $$x ^ { 2 } + g x + 3 g = 5$$ is formed.
3. Find the exact probability that the equation has no real roots.

3 The table shows the probability distribution of the random variable \(X\).

1. Explain why \(\mathrm { E } ( X ) = 25\).
2. Calculate \(\operatorname { Var } ( X )\).

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

1. Find the value of \(\mathrm { E } ( A )\).
2. Determine the value of \(\operatorname { Var } ( A )\).
3. The variable \(A\) represents the value in pence of a coin chosen at random from a pile. Mia picks one coin at random from the pile. She then adds, from a different source, another coin of the same value as the one that she has chosen, and one 50p coin.
   1. Find the mean of the value of the three coins.
   2. Find the variance of the value of the three coins.

1. The number of strokes, \(R\), taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.

   \(r\)	\(\leqslant 2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(\geqslant 9\)
   \(\mathrm{P}(R = r)\)	\(0\)	\(0.1\)	\(0.2\)	\(0.3\)	\(0.25\)	\(0.1\)	\(0.05\)	\(0\)

   1. Determine the probability that a member, selected at random, takes at least \(5\) strokes to complete the first hole. [1 mark]
   2. Calculate \(\mathrm{E}(R)\). [2 marks]
   3. Show that \(\mathrm{Var}(R) = 1.66\). [4 marks]
2. The number of strokes, \(S\), taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.

   \(s\)	\(\leqslant 2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(\geqslant 9\)
   \(\mathrm{P}(S = s)\)	\(0\)	\(0.15\)	\(0.4\)	\(0.3\)	\(0.1\)	\(0.03\)	\(0.02\)	\(0\)

   Assuming that \(R\) and \(S\) are independent:
   1. show that \(\mathrm{P}(R + S \leqslant 8) = 0.24\); [5 marks]
   2. calculate the probability that, when \(5\) members are selected at random, at least \(4\) of them complete the first two holes in fewer than \(9\) strokes; [3 marks]
   3. calculate \(\mathrm{P}(R = 4 \mid R + S \leqslant 8)\). [3 marks]

3 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k + 0.01 r ^ { 2 } \text { for } r = 1,2,3,4,5 .$$

1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table.
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 The number, \(X\), of children per family in a certain city is modelled by the probability distribution \(\mathrm { P } ( X = r ) = k ( 6 - r ) ( 1 + r )\) for \(r = 0,1,2,3,4\).

1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac { 1 } { 50 }\).

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(6 k\)	\(10 k\)

2. Calculate \(\mathrm { E } ( X )\).
3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children.

11 In this question you must show detailed reasoning. A biased four-sided spinner has edges numbered \(1,2,3,4\). When the spinner is spun, the probability that it will land on the edge numbered \(X\) is given by \(P ( X = x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 10 } x & x = 1,2,3,4 , \\ 0 & \text { otherwise } . \end{cases}\)

1. Draw a table showing the probability distribution of \(X\). The spinner is spun three times and the value of \(X\) is noted each time.
2. Find the probability that the third value of \(X\) is greater than the sum of the first two values of \(X\).

The probability distribution for the discrete random variable \(W\) is given in the table.

\(w\)	1	2	3	4
\(P(W = w)\)	0.25	0.36	\(x\)	\(x^2\)

1. Show that \(\text{Var}(W) = 0.8571\). [7]
2. Find \(\text{Var}(3W + 6)\). [1]

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]

2 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 2.3\) and \(\operatorname { Var } ( X ) = 3.01\), find the values of \(p , q\) and \(r\).

1. The discrete random variable \(X\) takes the values \(- 1,2,3,4\) and 7 only.

Given that $$\mathrm { P } ( X = x ) = \frac { 8 - x } { k } \text { for } x = - 1,2,3,4 \text { and } 7$$ find the value of \(\mathrm { E } ( X )\)

1 The table shows the probability distribution for a random variable X. Calculate \(\mathrm { E } ( \mathrm { X } )\) and \(\operatorname { Var } ( \mathrm { X } )\).

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

1. Find the mean of \(X\) in terms of \(k\). [2 marks]
2. Find the bias in using \((2\overline{X} - 5)\) as an estimator of \(k\). [3 marks]

Fifty observations of \(X\) were made giving a sample mean of 8.34 correct to 3 significant figures.

1. Calculate an unbiased estimate of \(k\). [2 marks]

1 The discrete random variable \(X\) has the following probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} \frac { 5 - x } { 10 } & x = 1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\) Circle your answer.
0.1
0.15
0.2
0.3

1 The discrete random variable \(X\) has the following probability distribution Find \(\mathrm { P } ( X > 18 )\) Circle your answer.
0.1
0.2
0.7
0.8

3 The possible values of the random variable \(X\) are the 8 integers in the set \(\{ - 2 , - 1,0,1,2,3,4,5 \}\). The probability of \(X\) being 0 is \(\frac { 1 } { 10 }\). The probabilities for all the other values of \(X\) are equal. Calculate

1. \(\mathrm { P } ( X < 2 )\),
2. the variance of \(X\),
3. the value of \(a\) for which \(\mathrm { P } ( - a \leqslant X \leqslant 2 a ) = \frac { 17 } { 35 }\).

4 A box contains 2 green sweets and 5 blue sweets. Two sweets are taken at random from the box, without replacement. The random variable \(X\) is the number of green sweets taken. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 Jasmine has one \(\\) 5\( coin, two \)\\( 2\) coins and two \(\\) 1\( coins. She selects two of these coins at random. The random variable \)X$ is the total value, in dollars, of these two coins.

1. Show that \(\mathrm { P } ( X = 7 ) = 0.2\).
2. Draw up the probability distribution table for \(X\).
3. Find the value of \(\operatorname { Var } ( X )\).

6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

3 A box contains 6 identical-sized discs, of which 4 are blue and 2 are red. Discs are taken at random from the box in turn and not replaced. Let \(X\) be the number of discs taken, up to and including the first blue one.

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 15 }\).
2. Draw up the probability distribution table for \(X\).

4 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable \(X\) represents the number of sites investigated. The probability distribution of \(X\) is shown below.

1. Find the expectation and variance of \(X\).
2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
3. Draw a suitable diagram to illustrate the distribution of \(X\).

5

1. Joshua plays a game in which he repeatedly tosses an unbiased coin. His game concludes when he obtains either a head or 5 tails in succession. The random variable \(N\) denotes the number of tosses of his coin required to conclude a game. By completing Table 3 below, calculate \(\mathrm { E } ( N )\).
2. Joshua's sister, Ruth, plays a separate game in which she repeatedly tosses a coin that is biased in such a way that the probability of a head in a single toss of her coin is \(\frac { 1 } { 4 }\). Her game also concludes when she obtains either a head or 5 tails in succession. The random variable \(M\) denotes the number of tosses of her coin required to conclude her game. Complete Table 4 below.
3. 1. Joshua and Ruth play their games simultaneously. Calculate the probability that Joshua and Ruth will conclude their games in an equal number of tosses of their coins.
   2. Joshua and Ruth play their games simultaneously on 3 occasions. Calculate the probability that, on at least 2 of these occasions, their games will be concluded in an equal number of tosses of their coins. Give your answer to three decimal places.
      (4 marks) \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 3}

      \(\boldsymbol { n }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)			\(\frac { 1 } { 8 }\)		\(\frac { 1 } { 16 }\)

      \end{table} \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Table 4}

      \(\boldsymbol { m }\)	1	2	3	4	5
      \(\mathbf { P } ( \boldsymbol { M } = \boldsymbol { m } )\)	\(\frac { 1 } { 4 }\)	\(\frac { 3 } { 16 }\)

      \end{table}

2 Three fair six-sided dice are thrown. The random variable \(X\) represents the highest of the three scores on the dice.

1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 91 } { 216 }\). The table shows the probability distribution of \(X\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 216 }\)	\(\frac { 7 } { 216 }\)	\(\frac { 19 } { 216 }\)	\(\frac { 37 } { 216 }\)	\(\frac { 61 } { 216 }\)	\(\frac { 91 } { 216 }\)

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table. The spinner is spun twice. The total of the two numbers on which it lands is denoted by \(X\).

1. Show that \(\mathrm { P } ( X = 2 ) = 0.18\). The probability distribution of \(X\) is given in the table.

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	0.49	0.28	0.18	0.04	0.01

2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 A fair tetrahedral die has four triangular faces, numbered \(1,2,3\) and 4 . The score when this die is thrown is the number on the face that the die lands on. This die is thrown three times. The random variable \(X\) is the sum of the three scores.

1. Show that \(\mathrm { P } ( X = 9 ) = \frac { 10 } { 64 }\).
2. Copy and complete the probability distribution table for \(X\).

   \(x\)	3	4	5	6	7	8	9	10	11	12
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 64 }\)			\(\frac { 12 } { 64 }\)

3. Event \(R\) is 'the sum of the three scores is 9 '. Event \(S\) is 'the product of the three scores is 16 '. Determine whether events \(R\) and \(S\) are independent, showing your working.

2 Two unbiased tetrahedral dice each have four faces numbered \(1,2,3\) and 4. The two dice are thrown together and the sum of the numbers on the faces on which they land is noted. Find the expected number of occasions on which this sum is 7 or more when the dice are thrown together 200 times.

The discrete random variable \(X\) has the probability function $$P(X = x) = \begin{cases} c(7 - 2x) & x = 0, 1, 2, 3 \\ k & x = 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(c\) and \(k\) are constants.

1. Show that \(16c + k = 1\) [2 marks]
2. Given that \(P(X \geq 3) = \frac{5}{8}\) find the value of \(c\) and the value of \(k\). [2 marks]

1. The discrete random variable \(X\) can only take the values \(1,2,3\) and 4 For these values the cumulative distribution function is defined by

$$\mathrm { F } ( x ) = k x ^ { 2 } \text { for } x = 1,2,3,4$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the probability distribution of \(X\).

The random variable \(X\) is defined as the difference (always positive or zero) between the scores when 2 ordinary dice are rolled.

1. Copy and complete the probability distribution table for \(X\). [2]

   \(x\)	0	1	2	3	4	5
   P(\(X = x\))

2. Find the expectation and variance of \(X\). [5]

6 The probability distribution for a random variable \(Y\) is shown in the table.

1. Calculate \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). Another random variable, \(Z\), is independent of \(Y\). The probability distribution for \(Z\) is shown in the table.

   \(z\)	1	2	3
   \(\mathrm { P } ( Z = z )\)	0.1	0.25	0.65

   One value of \(Y\) and one value of \(Z\) are chosen at random. Find the probability that
2. \(Y + Z = 3\),
3. \(Y \times Z\) is even.

A random variable \(X\) has probability distribution given by \(P(X = x) = \frac{1}{860}(1 + x)\) for \(x = 1, 2, 3, \ldots, 40\).

1. Find \(P(X > 39)\). [2]
2. Given that \(x\) is even, determine \(P(X < 10)\). [6]

The table below shows the probability distribution for a discrete random variable \(X\).

\(x\)	1	2	3	4	5
P(\(X = x\))	\(k\)	\(2k\)	\(4k\)	\(2k\)	\(k\)

Find the value of \(k\). Circle your answer. [1 mark] \(\frac{1}{2}\) \quad \(\frac{1}{4}\) \quad \(\frac{1}{10}\) \quad \(1\)

4 Martin has won a competition. For his prize he is given six sealed envelopes, of which he is allowed to open exactly three and keep their contents. Three of the envelopes each contain \(\pounds 5\) and the other three each contain \(\pounds 1000\). Since the envelopes are identical on the outside, he chooses three of them at random. Let \(\pounds X\) be the total amount of money that he receives in prize money.

1. Show that \(\mathrm { P } ( X = 15 ) = 0.05\). The probability distribution of \(X\) is given in the table below.

   \(r\)	15	1010	2005	3000
   \(\mathrm { P } ( X = r )\)	0.05	0.45	0.45	0.05

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 The table shows the probability distribution of a random variable \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Three values of \(X\) are chosen at random. Find the probability that \(X\) takes the value 2 at least twice.

The probability function of a discrete random variable \(X\) is given by $$p(x) = kx^2 \quad x = 1, 2, 3$$ where \(k\) is a positive constant.

1. Show that \(k = \frac{1}{14}\) [2]

Find

1. P\((X \geq 2)\) [2]
2. E\((X)\) [2]
3. Var\((1-X)\) [4]

1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 6.5\) Find \(\operatorname { Var } ( 4 X - 2 )\) Circle your answer.
2426102104

1 The discrete random variable \(X\) has probability distribution function $$\mathrm { P } ( X = x ) = \begin{cases} 0.45 & x = 1 \\ 0.25 & x = 2 \\ 0.25 & x = 3 \\ 0.05 & x = 4 \\ 0 & \text { otherwise } \end{cases}$$ State the mode of \(X\) Circle your answer.
0.25
0.45
1
2.5

2 Two fair six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 are thrown and the two scores are noted. The difference between the two scores is defined as follows.

• If the scores are equal the difference is zero.
• If the scores are not equal the difference is the larger score minus the smaller score.

Find the expectation of the difference between the two scores.

3 Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$

1. Verify that \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days.

1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 5\) Find \(\operatorname { Var } ( 4 X - 3 )\) Circle your answer.
17207780

9 The probability distribution of a random variable \(X\) is given in the table. Two values of \(X\) are chosen at random. Find the probability that the second value is greater than the first.

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]

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] \(a \operatorname { Var } ( X )\) \(a \operatorname { Var } ( X ) + b\) \(a ^ { 2 } \operatorname { Var } ( X )\) \(a ^ { 2 } \operatorname { Var } ( X ) + b\)

6 Three fair six-sided dice are thrown. The random variable \(X\) represents the highest of the three scores on the dice.

1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 91 } { 216 }\). The table shows the probability distribution of \(X\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 216 }\)	\(\frac { 7 } { 216 }\)	\(\frac { 19 } { 216 }\)	\(\frac { 37 } { 216 }\)	\(\frac { 61 } { 216 }\)	\(\frac { 91 } { 216 }\)

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1. The cumulative distribution function of the discrete random variable \(W\), which takes only the values 6,7 and 8 , is given by

$$F ( W ) = \frac { ( w + 3 ) ( w - 1 ) } { 77 } \text { for } w = 6,7,8$$ Find \(\mathrm { E } ( W )\)

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 \(\operatorname { Var } ( 2 - 3 Y )\)