2.04a Discrete probability distributions

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

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. \(\operatorname { Var } ( 7 X - 5 )\)

2. The discrete random variable \(X\) can take only the values 1,2 and 3 . For these values the cumulative distribution function is defined by $$\mathrm { F } ( x ) = \frac { x ^ { 3 } + k } { 40 } \quad x = 1,2,3$$

1. Show that \(k = 13\)
2. Find the probability distribution of \(X\). Given that \(\operatorname { Var } ( X ) = \frac { 259 } { 320 }\)
3. find the exact value of \(\operatorname { Var } ( 4 X - 5 )\).

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 discrete random variable \(X\) has the probability function

$$\mathrm { P } ( X = x ) = \begin{cases} k ( 1 - x ) ^ { 2 } & x = - 1,0,1 \text { and } 2 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 1 } { 6 }\)
2. Find \(\mathrm { E } ( X )\)
3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 3 }\)
4. Find \(\operatorname { Var } ( 1 - 3 X )\)

1. Sammy is studying the number of units of gas, \(g\), and the number of units of electricity, \(e\), used in her house each week. A random sample of 10 weeks use was recorded and the data for each week were coded so that \(x = \frac { g - 60 } { 4 }\) and \(y = \frac { e } { 10 }\). The results for the coded data are summarised below

$$\sum x = 48.0 \quad \sum y = 58.0 \quad \mathrm {~S} _ { x x } = 312.1 \quad \mathrm {~S} _ { y y } = 2.10 \quad \mathrm {~S} _ { x y } = 18.35$$

1. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). Give the values of \(a\) and \(b\) correct to 3 significant figures.
2. Hence find the equation of the regression line of \(e\) on \(g\) in the form \(e = c + d g\). Give the values of \(c\) and \(d\) correct to 2 significant figures.
3. Use your regression equation to estimate the number of units of electricity used in a week when 100 units of gas were used.
4. Find the probability distribution of \(X\) .
5. Write down the value of \(\mathrm { F } ( 1.8 )\) .
6. Find the probability distribution of \(X\) .勤

2.The discrete random variable \(X\) takes the values 1,2 and 3 and has cum
function \(\mathrm { F } ( x )\) given by \includegraphics[max width=\textwidth, alt={}, center]{4cf4f2d7-d912-4b65-a666-caa37009661a-04_24_37_182_2010}

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 \(\mathrm { P } ( X < 5 )\).

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 \(\operatorname { Var } ( a X + 3 ) = 53.4\)

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

$$\mathrm { P } ( X = x ) = \frac { 1 } { 10 } \quad x = 1,2,3 , \ldots 10$$

1. Write down the name given to this distribution.
2. Write down the value of
   1. \(\mathrm { P } ( X = 10 )\)
   2. \(\mathrm { P } ( X < 10 )\) The continuous random variable \(Y\) has the normal distribution \(\mathrm { N } \left( 10,2 ^ { 2 } \right)\)
3. Write down the value of
   1. \(\mathrm { P } ( Y = 10 )\)
   2. \(\mathrm { P } ( Y < 10 )\)

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.