5.02b Expectation and variance: discrete random variables

2. The random variable \(X\) has probability distribution

1. Given that \(\mathrm { E } ( X ) = 3.5\), write down two equations involving \(p\) and \(q\). Find
2. the value of \(p\) and the value of \(q\),
3. \(\operatorname { Var } ( X )\),
4. \(\operatorname { Var } ( 3 - 2 X )\).

1. The random variable \(X\) has probability function

$$\mathrm { P } ( X = x ) = \frac { ( 2 x - 1 ) } { 36 } \quad x = 1,2,3,4,5,6$$

1. Construct a table giving the probability distribution of \(X\). Find
2. \(\mathrm { P } ( 2 < X \leqslant 5 )\),
3. the exact value of \(\mathrm { E } ( X )\).
4. Show that \(\operatorname { Var } ( X ) = 1.97\) to 3 significant figures.
5. Find \(\operatorname { Var } ( 2 - 3 X )\).

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.

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

1. Show that \(k = 0.1\) Find
2. \(\mathrm { E } ( X )\)
3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. \(\operatorname { Var } ( 2 - 5 X )\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
5. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1\)
6. Complete the probability distribution table for \(X _ { 1 } + X _ { 2 }\)

   \(y\)	2	3	4	5	6	7	8
   \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)	0.01	0.04	0.10		0.25	0.24

7. Find \(\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)\)

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. A discrete random variable \(X\) takes only positive integer values. It has a cumulative distribution function \(\mathrm { F } ( x ) = \mathrm { P } ( X \leq x )\) defined in the table below.

1. Determine the probability function, \(\mathrm { P } ( X = x )\), of \(X\).
2. Calculate \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 5.76\).
3. Given that \(Y = 2 X + 3\), find the mean and variance of \(Y\).

3. A discrete random variable \(X\) has a probability function as shown in the table below, where \(a\) and \(b\) are constants. Given that \(\mathrm { E } ( X ) = 1.7\),

1. find the value of \(a\) and the value of \(b\). Find
2. \(\mathrm { P } ( 0 < X < 1.5 )\),
3. \(\mathrm { E } ( 2 X - 3 )\).
4. Show that \(\operatorname { Var } ( X ) = 1.41\).
5. Evaluate \(\operatorname { Var } ( 2 X - 3 )\).

5. The random variable \(X\) has probability function $$P ( X = x ) = \begin{cases} k x , & x = 1,2,3 \\ k ( x + 1 ) , & x = 4,5 \end{cases}$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the exact value of \(\mathrm { E } ( X )\).
3. Show that, to 3 significant figures, \(\operatorname { Var } ( X ) = 1.47\).
4. Find, to 1 decimal place, \(\operatorname { Var } ( 4 - 3 X )\).

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. \(\operatorname { Var } ( 4 - 3 X )\).
   

7. The random variable \(X\) has probability distribution

1. Given that \(\mathrm { E } ( X ) = 4.5\), write down two equations involving \(p\) and \(q\). Find
2. the value of \(p\) and the value of \(q\),
3. \(\mathrm { P } ( 4 < X \leqslant 7 )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 27.4\), find
4. \(\operatorname { Var } ( X )\),
5. \(\mathrm { E } ( 19 - 4 X )\),
6. \(\operatorname { Var } ( 19 - 4 X )\).

3. The random variable \(X\) has probability distribution given in the table below. Given that \(\mathrm { E } ( X ) = 0.55\), find

1. the value of \(p\) and the value of \(q\),
2. \(\operatorname { Var } ( X )\),
3. \(\mathrm { E } ( 2 X - 4 )\).

6. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } a ( 3 - x ) & x = 0,1,2 \\ b & x = 3 \end{array} \right.$$

1. Find \(\mathrm { P } ( X = 2 )\) and complete the table below.

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)	\(3 a\)	\(2 a\)		\(b\)

   Given that \(\mathrm { E } ( X ) = 1.6\)
2. Find the value of \(a\) and the value of \(b\). Find
3. \(\mathrm { P } ( 0.5 < X < 3 )\),
4. \(\mathrm { E } ( 3 X - 2 )\).
5. Show that the \(\operatorname { Var } ( X ) = 1.64\)
6. Calculate \(\operatorname { Var } ( 3 X - 2 )\).

3. The discrete random variable \(X\) has probability distribution given by where \(a\) is a constant.

1. Find the value of \(a\).
2. Write down \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
4. Find \(\operatorname { Var } ( Y )\).
5. Calculate \(\mathrm { P } ( X \geqslant Y )\).

1. A biased die with six faces is rolled. The discrete random variable \(X\) represents the score on the uppermost face. The probability distribution of \(X\) is shown in the table below.

1. Given that \(\mathrm { E } ( X ) = 4.2\) find the value of \(a\) and the value of \(b\).
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 20.4\)
3. Find \(\operatorname { Var } ( 5 - 3 X )\) A biased die with five faces is rolled. The discrete random variable \(Y\) represents the score which is uppermost. The cumulative distribution function of \(Y\) is shown in the table below.

   \(y\)	1	2	3	4	5
   \(\mathrm {~F} ( y )\)	\(\frac { 1 } { 10 }\)	\(\frac { 2 } { 10 }\)	\(3 k\)	\(4 k\)	\(5 k\)

4. Find the value of \(k\).
5. Find the probability distribution of \(Y\). Each die is rolled once. The scores on the two dice are independent.
6. Find the probability that the sum of the two scores equals 2
   

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

1. Show that \(p = 0.2\) Find
2. \(\mathrm { E } ( X )\)
3. \(\mathrm { F } ( 0 )\)
4. \(\mathrm { P } ( 3 X + 2 > 5 )\) Given that \(\operatorname { Var } ( X ) = 13.35\)
5. find the possible values of \(a\) such that \(\operatorname { Var } ( a X + 3 ) = 53.4\)

5. The discrete random variable \(X\) has the probability function $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 2,4,6 \\ k ( x - 2 ) & x = 8 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 18 }\)
2. Find the exact value of \(\mathrm { F } ( 5 )\).
3. Find the exact value of \(\mathrm { E } ( X )\).
4. Find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\).
5. Calculate \(\operatorname { Var } ( 3 - 4 X )\) giving your answer to 3 significant figures.

1. In a quiz, a team gains 10 points for every question it answers correctly and loses 5 points for every question it does not answer correctly. The probability of answering a question correctly is 0.6 for each question. One round of the quiz consists of 3 questions.

The discrete random variable \(X\) represents the total number of points scored in one round. The table shows the incomplete probability distribution of \(X\)

1. Show that the probability of scoring 15 points in a round is 0.432
2. Find the probability of scoring 0 points in a round.
3. Find the probability of scoring a total of 30 points in 2 rounds.
4. Find \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\) In a bonus round of 3 questions, a team gains 20 points for every question it answers correctly and loses 5 points for every question it does not answer correctly.
6. Find the expected number of points scored in the bonus round.
   

2. The discrete random variable \(X\) has the following probability distribution, where \(p\) and \(q\) are constants.

1. Write down an equation in \(p\) and \(q\) Given that \(\mathrm { E } ( X ) = 0.4\)
2. find the value of \(q\)
3. hence find the value of \(p\) Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.275\)
4. find \(\operatorname { Var } ( X )\) Sarah and Rebecca play a game.
   A computer selects a single value of \(X\) using the probability distribution above.
   Sarah's score is given by the random variable \(S = X\) and Rebecca's score is given by the random variable \(R = \frac { 1 } { X }\)
5. Find \(\mathrm { E } ( R )\) Sarah and Rebecca work out their scores and the person with the higher score is the winner. If the scores are the same, the game is a draw.
6. Find the probability that
   1. Sarah is the winner,
   2. Rebecca is the winner.

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 \(\mathrm { P } \left( X ^ { 2 } = 1 \right)\).
   \section*{} \section*{
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   } \(T\)

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

5. The score when a spinner is spun is given by the discrete random variable \(X\) with the following probability distribution, where \(a\) and \(b\) are probabilities.

1. Explain why \(\mathrm { E } ( X ) = 2\)
2. Find a linear equation in \(a\) and \(b\). Given that \(\operatorname { Var } ( X ) = 7.1\)
3. find a second equation in \(a\) and \(b\) and simplify your answer.
4. Solve your two equations to find the value of \(a\) and the value of \(b\). The discrete random variable \(Y = 10 - 3 X\)
5. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\) The spinner is spun once.
6. Find \(\mathrm { P } ( Y > X )\).

4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .

1. Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
   (2 marks)
   The random variable \(A\) represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
2. Find the probability distribution of \(A\).
3. Calculate the probability that the customer types in the correct number in four or fewer attempts.
4. Calculate \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
5. Find \(\mathrm { F } ( 1 + \mathrm { E } ( A ) )\).