Calculate Var(X) from table

3 The table shows the probability distribution of the random variable \(X\).

1. Explain why \(\mathrm { E } ( X ) = 25\).
2. Calculate \(\operatorname { Var } ( X )\).

3 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.

1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
2. Draw up the probability distribution table for \(X\).
3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).

7 The probability distribution of a discrete random variable, \(X\), is shown below.

1. Find \(\mathrm { E } ( X )\) in terms of \(a\).
2. Show that \(\operatorname { Var } ( X ) = 4 a ( 1 - a )\).

2

1. The probability distribution of a random variable \(W\) is shown in the table.

   \(w\)	0	2	4
   \(\mathrm { P } ( W = w )\)	0.3	0.4	0.3

   Calculate \(\operatorname { Var } ( W )\).
2. The random variable \(X\) has probability distribution given by $$\mathrm { P } ( X = x ) = k ( x + 1 ) \quad \text { for } x = 1,2,3,4 .$$
   1. Show that \(k = \frac { 1 } { 14 }\).
   2. Calculate \(\mathrm { E } ( X )\).

4 The table shows the probability distribution of the random variable \(X\).

1. Explain why \(\mathrm { E } ( X ) = 25\).
2. Calculate \(\operatorname { Var } ( X )\).

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

1. Find the value of \(\mathrm { E } ( A )\).
2. Determine the value of \(\operatorname { Var } ( A )\).
3. The variable \(A\) represents the value in pence of a coin chosen at random from a pile. Mia picks one coin at random from the pile. She then adds, from a different source, another coin of the same value as the one that she has chosen, and one 50p coin.
   1. Find the mean of the value of the three coins.
   2. Find the variance of the value of the three coins.

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. A red spinner is designed so that the score \(R\) is given by the following probability distribution.

1. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 15.8\) Given also that \(\mathrm { E } ( R ) = 3.7\)
2. find the standard deviation of \(R\), giving your answer to 2 decimal places. A yellow spinner is designed so that the score \(Y\) is given by the probability distribution in the table below. The cumulative distribution function \(\mathrm { F } ( y )\) is also given.

   \(y\)	2	3	4	5	6
   \(\mathrm { P } ( Y = y )\)	0.1	0.2	0.1	\(a\)	\(b\)
   \(\mathrm {~F} ( y )\)	0.1	0.3	0.4	\(c\)	\(d\)

3. Write down the value of \(d\) Given that \(\mathrm { E } ( Y ) = 4.55\)
4. find the value of \(c\) Pabel and Jessie play a game with these two spinners.
   Pabel uses the red spinner.
   Jessie uses the yellow spinner.
   They take turns to spin their spinner.
   The winner is the first person whose spinner lands on the number 2 and the game ends. Jessie spins her spinner first.
5. Find the probability that Jessie wins on her second spin.
6. Calculate the probability that, in a game, the score on Pabel's first spin is the same as the score on Jessie's first spin.

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 \(\mathrm { P } ( X < 3 \mid Y < 20 )\)

1. 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 \(\mathrm { P } ( X \geqslant Y )\).

5 The Globe Express agency organises trips to the theatre. The cost, \(\pounds X\), of these trips can be modelled by the following probability distribution:

1. Calculate the mean and standard deviation of \(X\).
2. For special celebrity charity performances, Globe Express increases the cost of the trips to \(\pounds Y\), where $$Y = 10 X + 250$$ Determine the mean and standard deviation of \(Y\).

5 Aiden takes his car to a garage for its MOT test. The probability that his car will need to have \(X\) tyres replaced is shown in the table.

1. Show that the mean of \(X\) is 1.85 and calculate the variance of \(X\).
2. The charge for the MOT test is \(\pounds c\) and the cost of each new tyre is \(\pounds n\). The total amount that Aiden must pay the garage is \(\pounds T\).
   1. Express \(T\) in terms of \(c , n\) and \(X\).
   2. Hence, using your results from part (a), find expressions for \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).

5 The discrete random variable \(R\) has the following probability distribution.

1. Calculate exact values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
2. 1. By tabulating the probability distribution for \(X = \frac { 1 } { R ^ { 2 } }\), show that \(\mathrm { E } ( X ) = \frac { 25 } { 64 }\).
   2. Hence find the value of the mean of the area of a rectangle which has sides of length \(\frac { 8 } { R }\) and \(\left( R + \frac { 8 } { R } \right)\).
      (3 marks)

A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.

\(x\)	0	1	2	3
P\((X = x)\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(\frac{1}{8}\)	\(\frac{1}{8}\)

1. Calculate E\((X)\) and Var\((X)\). [5]

The die is thrown 10 times.

1. Find the probability that there are not more than 4 throws on which the score is 1. [2]
2. Find the probability that there are exactly 4 throws on which the score is 2. [3]

In her purse, Katharine has two £5 notes, two £10 notes and one £20 note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(£X\).

1. 1. Show that P(X = 10) = 0.1. [1]
   2. Show that P(X = 30) = 0.2. [2]
   The table shows the probability distribution of X.

   \(r\)	10	15	20	25	30
   P(X = r)	0.1	0.4	0.1	0.2	0.2

2. Find E(X) and Var(X). [5]

Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.

\(x\)	0	1	2	3	4	5	6
P(X = x)	0	0.19	0.26	0.20	0.13	0.07	0.15

1. Find the probability that a member borrows more than 3 books. [1 mark]
2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books. [3 marks]
3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\). [4 marks]
4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\). [2 marks]
5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
   1. the mean amount that will be paid by a member;
   2. the standard deviation of the amount that will be paid by a member.
   [3 marks]