Calculate E(X) from given distribution

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

2 Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).

1. Draw a vertical line chart to illustrate the probability distribution.
2. Use a probability argument to show that
   (A) \(\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }\),
   (B) \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 6 }\).
3. Find the mean value of \(X\).

1 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.

1. Explain why the case \(X = 3\) is impossible.
2. Explain why \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }\).
3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3. A discrete random variable \(X\) has the probability function shown in the table below. Find

1. \(\mathrm { P } ( 1 < X \leq 3 )\),
2. \(\mathrm { F } ( 2.6 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } ( 2 X - 3 )\),
5. \(\operatorname { Var } ( X )\)

4 Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).

1. Draw a vertical line chart to illustrate the probability distribution.
2. Use a probability argument to show that
   (A) \(\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }\),
   (B) \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 6 }\).
3. Find the mean value of \(X\).

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 \(\mathrm { E } ( X )\)

7. The score \(S\) when a spinner is spun has the following probability distribution.

1. Find \(\mathrm { E } ( S )\).
2. Show that \(\mathrm { E } \left( S ^ { 2 } \right) = 10.4\)
3. Hence find \(\operatorname { Var } ( S )\).
4. Find
   1. \(\mathrm { E } ( 5 S - 3 )\),
   2. \(\operatorname { Var } ( 5 S - 3 )\).
5. Find \(\mathrm { P } ( 5 S - 3 > S + 3 )\) The spinner is spun twice.
   The score from the first spin is \(S _ { 1 }\) and the score from the second spin is \(S _ { 2 }\)
   The random variables \(S _ { 1 }\) and \(S _ { 2 }\) are independent and the random variable \(X = S _ { 1 } \times S _ { 2 }\)
6. Show that \(\mathrm { P } \left( \left\{ S _ { 1 } = 1 \right\} \cap X < 5 \right) = 0.16\)
7. Find \(\mathrm { P } ( X < 5 )\).

6. The score, \(X\), for a biased spinner is given by the probability distribution Find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\) A biased coin has one face labelled 2 and the other face labelled 5 The score, \(Y\), when the coin is spun has $$\mathrm { P } ( Y = 5 ) = p \quad \text { and } \quad \mathrm { E } ( Y ) = 3$$
3. Form a linear equation in \(p\) and show that \(p = \frac { 1 } { 3 }\)
4. Write down the probability distribution of \(Y\). Sam plays a game with the spinner and the coin.
   Each is spun once and Sam calculates his score, \(S\), as follows $$\begin{aligned} & \text { if } X = 0 \text { then } S = Y ^ { 2 }
   & \text { if } X \neq 0 \text { then } S = X Y \end{aligned}$$
5. Show that \(\mathrm { P } ( S = 30 ) = \frac { 1 } { 12 }\)
6. Find the probability distribution of \(S\).
7. Find \(\mathrm { E } ( S )\). Charlotte also plays the game with the spinner and the coin.
   Each is spun once and Charlotte ignores the score on the coin and just uses \(X ^ { 2 }\) as her score. Sam and Charlotte each play the game a large number of times.
8. State, giving a reason, which of Sam and Charlotte should achieve the higher total score.

   END

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

Find

1. \(\mathrm { F } ( 0.5 )\),
2. \(\mathrm { P } \left( { } ^ { - } 1 < Y < 1.9 \right)\),
3. \(\mathrm { E } ( Y )\),
4. \(\mathrm { E } ( 3 Y - 1 )\).

4

1. A red biased tetrahedral die is rolled. The number, \(X\), on the face on which it lands has the probability distribution given by

   \(\boldsymbol { x }\)	1	2	3	4
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.2	0.1	0.4	0.3

   1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   2. The red die is now rolled three times. The random variable \(S\) is the sum of the three numbers obtained. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
2. A blue biased tetrahedral die is rolled. The number, \(Y\), on the face on which it lands has the probability distribution given by $$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3
   \frac { 7 } { 10 } & y = 4 \end{cases}$$ The random variable \(T\) is the value obtained when the number on the face on which it lands is multiplied by 3 . Calculate \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
3. Calculate:
   1. \(\mathrm { P } ( X > 1 )\);
   2. \(\mathrm { P } ( X + T \leqslant 9\) and \(X > 1 )\);
   3. \(\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )\).

3 Morecrest football team always scores at least one goal but never scores more than four goals in each game. The number of goals, \(R\), scored in each game by the team can be modelled by the following probability distribution.

1. Calculate exact values for the mean and variance of \(R\).
2. Next season the team will play 32 games. They expect to win \(90 \%\) of the games in which they score at least three goals, half of the games in which they score exactly two goals and \(20 \%\) of the games in which they score exactly one goal. Find, for next season:
   1. the number of games in which they expect to score at least three goals;
   2. the number of games that they expect to win.

7

1. The number of text messages, \(N\), sent by Peter each month on his mobile phone never exceeds 40. When \(0 \leqslant N \leqslant 10\), he is charged for 5 messages.
   When \(10 < N \leqslant 20\), he is charged for 15 messages.
   When \(20 < N \leqslant 30\), he is charged for 25 messages.
   When \(30 < N \leqslant 40\), he is charged for 35 messages.
   The number of text messages, \(Y\), that Peter is charged for each month has the following probability distribution:

   \(\boldsymbol { y }\)	5	15	25	35
   \(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	0.1	0.2	0.3	0.4

   1. Calculate the mean and the standard deviation of \(Y\).
   2. The Goodtime phone company makes a total charge for text messages, \(C\) pence, each month given by $$C = 10 Y + 5$$ Calculate \(\mathrm { E } ( C )\).
2. The number of text messages, \(X\), sent by Joanne each month on her mobile phone is such that $$\mathrm { E } ( X ) = 8.35 \quad \text { and } \quad \mathrm { E } \left( X ^ { 2 } \right) = 75.25$$ The Newtime phone company makes a total charge for text messages, \(T\) pence, each month given by $$T = 0.4 X + 250$$ Calculate \(\operatorname { Var } ( T )\).

6

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.

   \(\boldsymbol { r }\)	\(\leqslant 2\)	3	4	5	6	7	8	\(\geqslant 9\)
   \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	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.
   2. Calculate \(\mathrm { E } ( R )\).
   3. Show that \(\operatorname { Var } ( R ) = 1.66\).
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.

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

   Assuming that \(R\) and \(S\) are independent:
   1. show that \(\mathrm { P } ( R + S \leqslant 8 ) = 0.24\);
   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. calculate \(\mathrm { P } ( R = 4 \mid R + S \leqslant 8 )\).
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-15_2484_1709_223_153}
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-16_2484_1712_223_153}
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-17_2484_1709_223_153}

2 In a quiz, competitors have to match 5 landmarks to the 5 British counties which the landmarks are in. The random variable \(X\) represents the number of correct matches that a competitor gets, assuming that the competitor guesses randomly. The probability distribution of \(X\) is given in the following table.

1. Explain why \(\mathrm { P } ( X = 4 )\) must be 0 .
2. Explain how the value \(\frac { 1 } { 120 }\) for \(\mathrm { P } ( X = 5 )\) is calculated.
3. Draw a graph to illustrate the distribution.
4. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
   • There are 12 competitors in the quiz. Assuming that they all guess randomly, find the probability that at least one of them gets all five matches correct.

1 The random variable \(X\) represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of \(X\) is given in the table.

1. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   On average 65\% of eggs laid result in a young bird successfully leaving the nest.
2. 1. Find the mean number of young birds that successfully leave the nest.
   2. Find the standard deviation of the number of young birds that successfully leave the nest.

2 The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.

1. (A) Explain why \(\mathrm { E } ( X ) = 2\).
   (B) Find \(\operatorname { Var } ( X )\).
2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250 X - 80\).
   Find
   • \(\mathrm { E } ( Y )\) and
   • \(\operatorname { Var } ( Y )\).
   The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.
3. Explain why it would be inappropriate to test all the remote controls in this way.
4. State an advantage of using random sampling in this context. A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between 0 and 10 (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.
5. Let \(X\) be the number of points gained after shopping once. Find
   • the mean of \(X\)
   • the variance of \(X\).
   • Let \(Y\) be the number of points gained after shopping twice.
   Find
   • the mean of \(Y\)
   • the variance of \(Y\).
   • Find the probability of the most likely number of points gained after shopping twice. Justify your answer.
   • State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day.
   Jane records the number of junk emails which she receives each day. During working hours (9am to 5pm, Monday to Friday) the mean number of junk emails is 7.4 per day. Outside working hours ( 5 pm to 9am), the mean number of junk emails is 0.3 per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".
6. Find the probability that the number of junk emails which she receives between 9am and 5pm on a Monday is
   (A) exactly 10 ,
   (B) at least 10 .
7. (A) What assumption must you make to calculate the probability that the number of junk emails which she receives from 9am Monday to 9am Tuesday is at most 20?
   (B) Find the probability.

1 The number of insurance policy sales made per month by a salesperson is modelled by the random variable \(X\), with probability distribution shown in the table.

1. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   The salesperson is paid a basic salary of \(\pounds 1000\) per month plus \(\pounds 500\) for each policy that is sold.
2. Find the mean and standard deviation of the salesperson's monthly salary.

1. The random variable \(X\) represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of \(X\) is given in the table.

1. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   On average \(65 \%\) of eggs laid result in a young bird successfully leaving the nest.
2. 1. Find the mean number of young birds that successfully leave the nest.
   2. Find the standard deviation of the number of young birds that successfully leave the nest.
      [0pt] [BLANK PAGE]