5.02a Discrete probability distributions: general

4. A discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } k ( 2 - x ) & x = 0,1 \\ k ( 3 - x ) & x = 2,3 \\ k ( x + 1 ) & x = 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 9 }\) Find the exact value of
2. \(\mathrm { P } ( 1 \leqslant X < 4 )\)
3. \(\mathrm { E } ( X )\)
4. \(\mathrm { E } \left( X ^ { 2 } \right)\)
5. \(\operatorname { Var } ( 3 X + 1 )\)

1. The discrete random variable \(X\) represents the score when a biased spinner is spun. The probability distribution of \(X\) is given by

where \(p\) and \(q\) are probabilities.

1. Find \(\mathrm { E } ( X )\). Given that \(\operatorname { Var } ( X ) = 2.5\)
2. find the value of \(p\).
3. Hence find the value of \(q\). Amar is invited to play a game with the spinner.
   The spinner is spun once and \(X _ { 1 }\) is the score on the spinner. If \(X _ { 1 } > 0\) Amar wins the game.
   If \(X _ { 1 } = 0\) Amar loses the game.
   If \(X _ { 1 } < 0\) the spinner is spun again and \(X _ { 2 }\) is the score on this second spin and if \(X _ { 1 } + X _ { 2 } > 0\) Amar wins the game, otherwise Amar loses the game.
4. Find the probability that Amar wins the game. Amar does not want to lose the game.
   He says that because \(\mathrm { E } ( X ) > 0\) he will play the game.
5. State, giving a reason, whether or not you would agree with Amar.

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

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.
7. Find the probability of achieving this value. \includegraphics[max width=\textwidth, alt={}, center]{81d5e460-9559-4d25-aa08-6440559aec83-21_2255_50_314_34}

1. The discrete random variable \(X\) is defined by the cumulative distribution function

where \(k\) is a constant.

1. Find the probability distribution of \(X\).
2. Find \(\mathrm { P } ( 1.5 < X \leqslant 3.5 )\) The random variable \(Y = 12 - 7 X\)
3. Calculate Var(Y)
4. Calculate \(\mathrm { P } ( 4 X \leqslant | Y | )\)

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. Adana selects one number at random from the distribution of \(X\) which has the following probability distribution.

1. Given that the number selected by Adana is not 5 , write down the probability it is 0
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 75\)
3. Find \(\operatorname { Var } ( X )\)
4. Find \(\operatorname { Var } ( 4 - 3 X )\) Bruno and Charlie each independently select one number at random from the distribution of \(X\)
5. Find the probability that the number Bruno selects is greater than the number Charlie selects. Devika multiplies Bruno's number by Charlie's number to obtain a product, \(D\)
6. Determine the probability distribution of \(D\)

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

1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 2 } { 5 }\)
2. Find \(\operatorname { Var } \left( \frac { 1 } { X } \right)\) The random variable \(Y = \frac { 30 } { X }\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } ( X < 3 \mid Y < 20 )\)

7. Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered \(0,1,2\), and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable \(R\) is the score on the red die and the random variable \(B\) is the score on the blue die.

1. Find \(\mathrm { P } ( R = 3\) and \(B = 0 )\). The random variable \(T\) is \(R\) multiplied by \(B\).
2. Complete the diagram below to represent the sample space that shows all the possible values of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-13_732_771_834_621} \section*{Sample space diagram of \(T\)}
3. The table below represents the probability distribution of the random variable \(T\).

   \(t\)	0	1	2	3	4	6	9
   \(\mathrm { P } ( T = t )\)	\(a\)	\(b\)	\(1 / 8\)	\(1 / 8\)	\(c\)	\(1 / 8\)	\(d\)

   Find the values of \(a , b , c\) and \(d\). Find the values of
4. \(\mathrm { E } ( T )\),
5. \(\operatorname { Var } ( T )\).

3. When Rohit plays a game, the number of points he receives is given by the discrete random variable \(X\) with the following probability distribution.

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { F } ( 1.5 )\).
3. Show that \(\operatorname { Var } ( X ) = 1\)
4. Find \(\operatorname { Var } ( 5 - 3 X )\). Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.
5. Find the probability that Rohit wins the prize.

3. The discrete random variable \(X\) can take only the values \(2,3,4\) or 6 . For these values the probability distribution function is given by where \(k\) is a positive integer.

1. Show that \(k = 3\) Find
2. \(\mathrm { F } ( 3 )\)
3. \(\mathrm { E } ( X )\)
4. \(\mathrm { E } \left( X ^ { 2 } \right)\)
5. \(\operatorname { Var } ( 7 X - 5 )\)

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.

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

1. \(\alpha\),
2. \(\mathrm { P } ( - 1 < X \leq 2 )\),
3. \(\mathrm { F } ( - 0.4 )\),
4. \(\mathrm { E } ( 3 X + 4 )\),
5. \(\operatorname { Var } ( 2 X + 3 )\).

4. The discrete random variable \(X\) has probability distribution The cumulative distribution function of \(X\) is given by

1. Find the values of \(a , b , c , d\) and \(e\).
2. Write down the value of \(\mathrm { P } \left( X ^ { 2 } = 1 \right)\).
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   } \(T\)

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

where \(a , b\) and \(c\) are probabilities.
The cumulative distribution function of \(X\) is \(\mathrm { F } ( x )\) and \(\mathrm { F } ( 3 ) = 0.2\) and \(\mathrm { F } ( 6 ) = 0.8\)

1. Find the value of \(a\), the value of \(b\) and the value of \(c\).
2. Write down the value of \(\mathrm { F } ( 7 )\).
   

2. A fairground game involves trying to hit a moving target with a gunshot. A round consists of up to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of one another.

1. Find the probability that Linda scores 30 points in a round. The random variable \(X\) is the number of points Linda scores in a round.
2. Find the probability distribution of \(X\).
3. Find the mean and the standard deviation of \(X\). A game consists of 2 rounds.
4. Find the probability that Linda scores more points in round 2 than in round 1.

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. \(\operatorname { Var } ( X )\).

4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } 0.2 , & x = - 3 , - 2 \\ \alpha , & x = - 1,0 \\ 0.1 , & x = 1,2 . \end{array}$$ Find

1. \(\alpha\),
2. \(\mathrm { P } ( - 1 \leq X < 2 )\),
3. \(\mathrm { F } ( 0.6 )\),
4. the value of \(a\) such that \(\mathrm { E } ( a X + 3 ) = 1.2\),
5. \(\operatorname { Var } ( X )\),
6. \(\operatorname { Var } ( 3 X - 2 )\).

2. A farmer sells boxes of eggs. The eggs are sold in boxes of 6 eggs and boxes of 12 eggs in the ratio \(n : 1\) A random sample of three boxes is taken.
The number of eggs in the first box is denoted by \(X _ { 1 }\) The number of eggs in the second box is denoted by \(X _ { 2 }\) The number of eggs in the third box is denoted by \(X _ { 3 }\) The random variable \(T = X _ { 1 } + X _ { 2 } + X _ { 3 }\) Given that \(\mathrm { P } ( T = 18 ) = 0.729\)

1. show that \(n = 9\)
2. find the sampling distribution of \(T\) The random variable \(R\) is the range of \(X _ { 1 } , X _ { 2 } , X _ { 3 }\)
3. Using your answer to part (b), or otherwise, find the sampling distribution of \(R\)

6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.

1. Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
2. Suggest a suitable sampling frame.
3. Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
   When a ball is selected at random, the random variable \(X\) represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable \(M\) is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
4. show that \(\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }\)

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5. A bag contains a large number of counters with \(35 \%\) of the counters having a value of 6 and \(65 \%\) of the counters having a value of 9 A random sample of size 2 is taken from the bag and the value of each counter is recorded as \(X _ { 1 }\) and \(X _ { 2 }\) respectively. The statistic \(Y\) is calculated using the formula $$Y = \frac { 2 X _ { 1 } + X _ { 2 } } { 3 }$$

1. List all the possible values of \(Y\).
2. Find the sampling distribution of \(Y\).
3. Find \(\mathrm { E } ( Y )\).

5. A bag contains a large number of coins. It contains only \(1 \mathrm { p } , 5 \mathrm { p }\) and 10 p coins. The fraction of 1 p coins in the bag is \(q\), the fraction of 5 p coins in the bag is \(r\) and the fraction of 10p coins in the bag is \(s\). Two coins are selected at random from the bag and the coin with the highest value is recorded. Let \(M\) represent the value of the highest coin. The sampling distribution of \(M\) is given below

1. List all the possible samples of two coins which may be selected.
2. Find the value of \(q\), the value of \(r\) and the value of \(s\)

6. At a men's tennis tournament there are 3 , 4 or 5 sets in a match. Over many years, data collected show that 50\% of matches last for exactly 3 sets, 30\% of matches last for exactly 4 sets and 20\% of matches last for exactly 5 sets. A random sample of 3 tennis matches is taken. The number of sets in each match is recorded as \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\) respectively. The random variable \(M\) represents the maximum value of \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\)

1. List all the samples where \(M \neq 5\)
2. Find the sampling distribution of \(M\)
3. Write down the mode of \(S _ { 1 }\) and the mode of \(M\)

6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables: The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).

1. Tabulate the probability distribution for \(Z\).
2. Calculate \(\mathrm { E } ( Z )\).
3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
4. Calculate Var (3Z-4).

1. Two spinners are in the form of an equilateral triangle, whose three regions are labelled 1,2 and 3, and a square, whose four regions are labelled \(1,2,3\) and 4 . Both spinners are biased and the probability distributions for the scores \(X\) and \(Y\) obtained when they are spun are respectively:

1. Find the values of \(p\) and \(q\).
2. Find the probability that, when the two spinners are spun together, the sum of the two scores is (i) 5, (ii) less than 4 .
3. State an assumption that you have made in answering part (b) and explain why it is likely to be justifiable.
4. Calculate \(\mathrm { E } ( X + Y )\).