Probability distribution from formula

5 In a particular discrete probability distribution the random variable \(X\) takes the value \(\frac { 120 } { r }\) with probability \(\frac { r } { 45 }\), where \(r\) takes all integer values from 1 to 9 inclusive.

1. Show that \(\mathrm { P } ( X = 40 ) = \frac { 1 } { 15 }\).
2. Construct the probability distribution table for \(X\).
3. Which is the modal value of \(X\) ?
4. Find the probability that \(X\) lies between 18 and 100 .

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.

4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6 .$$

1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(k\)					\(11 k\)

2. Find the mean of \(X\). A fair six-sided die is rolled three times.
3. Find the probability that the total score is 16 .

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

12 The discrete random variable \(X\) takes values 1, 2, 3, 4 and 5, and its probability distribution is defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 1 , \\ \frac { 1 } { 2 } \mathrm { P } ( X = x - 1 ) & x = 2,3,4,5 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac { 16 } { 31 }\). The discrete probability distribution for \(X\) is given in the table.

   \(x\)	1	2	3	4	5
   \(\mathrm { P } ( X = x )\)	\(\frac { 16 } { 31 }\)	\(\frac { 8 } { 31 }\)	\(\frac { 4 } { 31 }\)	\(\frac { 2 } { 31 }\)	\(\frac { 1 } { 31 }\)

2. Find the probability that \(X\) is odd. Two independent values of \(X\) are chosen, and their sum \(S\) is found.
3. Find the probability that \(S\) is odd.
4. Find the probability that \(S\) is greater than 8 , given that \(S\) is odd. Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm { P } ( Y = y + 1 ) = \frac { 1 } { 2 } \mathrm { P } ( Y = y ) \quad \text { for all positive integers } y .$$
5. Find \(\mathrm { P } ( Y = 1 )\).
6. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car.

14 The probability distribution of a random variable \(X\) is modelled as follows. \(\mathrm { P } ( X = x ) = \begin{cases} \frac { k } { x } & x = 1,2,3,4 , \\ 0 & \text { otherwise, } \end{cases}\) where \(k\) is a constant.

1. Show that \(k = \frac { 12 } { 25 }\).
2. Show in a table the values of \(X\) and their probabilities.
3. The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\). Find \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } + X _ { 3 } \right)\). In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7 .
4. Determine the probability that a total of exactly 7 is first reached on the 5th observation. \section*{OCR} Oxford Cambridge and RSA

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. \(\operatorname { Var } ( 3 X + 1 )\)

2. The discrete random variable \(X\) has the probability function shown below. $$P ( X = x ) = \left\{ \begin{array} { l c } \frac { k } { x } , & x = 1,2,3,4 \\ 0 , & \text { otherwise } . \end{array} \right.$$

1. Show that \(k = \frac { 12 } { 25 }\) Find
2. \(\mathrm { F } ( 2 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } \left( X ^ { 2 } + 2 \right)\).

4 A house has a total of five bedrooms, at least one of which is always rented.
The probability distribution for \(R\), the number of bedrooms that are rented at any given time, is given by $$\mathrm { P } ( R = r ) = \begin{cases} 0.5 & r = 1 \\ 0.4 ( 0.6 ) ^ { r - 1 } & r = 2,3,4 \\ 0.0296 & r = 5 \end{cases}$$

1. Complete the table below.
2. Find the probability that fewer than 3 bedrooms are not rented at any given time.
3. 1. Find the value of \(\mathrm { E } ( R )\).
   2. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 4.8784\) and hence find the value of \(\operatorname { Var } ( R )\).
4. Bedrooms are rented on a monthly basis. The monthly income, \(\pounds M\), from renting bedrooms in the house may be modelled by $$M = 1250 R - 282$$ Find the mean and the standard deviation of \(M\).

   \(\boldsymbol { r }\)	1	2	3	4	5
   \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.5				0.0296

5 In a computer game, players try to collect five treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable \(X\). The probability distribution of \(X\) is defined by $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 2 } & x = 1,2,3,4 \\ k & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. 1. Show that \(k = \frac { 1 } { 20 }\).
   2. Calculate the value of \(\mathrm { E } ( X )\).
   3. Show that \(\operatorname { Var } ( X ) = 1.5275\).
   4. Find the probability that Isaac collects more than 2 treasures.
2. The number of points that Isaac scores for collecting treasures is \(Y\) where $$Y = 100 X - 50$$ Calculate the mean and the standard deviation of \(Y\).

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 \(X\).

A regular tetrahedron has its faces numbered 1, 2, 3 and 4. It is weighted so that when it is thrown, the probability of each face being in contact with the table is inversely proportional to the number on that face. This number is represented by the random variable \(X\).

1. Show that \(P(X = 1) = \frac{12}{25}\) and find the probabilities of the other values of \(X\). [5 marks]
2. Calculate the mean and the variance of \(X\). [5 marks]

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

1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table. [3]
2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). [5]

A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.

1. Write down the probability distribution for the random variable, \(X\), the score on a single throw of the die. [4]
2. Show that E\((X) = \frac{33}{8}\). [3]
3. Find E\((4X - 1)\). [2]
4. Find Var\((X)\). [4]

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 P(\(X = r\)) = \(\frac{1}{40}r(r + 1)\) for \(r = 1, 2, 3, 4\).

1. Verify that P(\(X = 4\)) = \(\frac{1}{2}\). [1]
2. Calculate E(\(X\)) and Var(\(X\)). [5]
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days. [2]

The number, \(X\), of children per family in a certain city is modelled by the probability distribution 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 \(\frac{1}{50}\). [3]

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

2. Calculate E(\(X\)). [2]
3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children. [1]