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 b clb sed 6 p p rb ck ad 2 h r ck b to Mrs Ho . Sb cb es 4 6 tb se b at rach to take with b r o b id y. Th rach \& riable \(X\) rep esen s tb m br \(\mathbf { b }\) p \(\mathbf { p }\) rb ck b sh cb es.

1. Sth that th p b b lityt \(\mathbf { h }\) tsb cb es extlye perb clb is \(\frac { 3 } { 14 }\). [R
2. Draw up b pb b lityd strib in tab e fo \(X\).
3. Yu reg it h t \(\mathrm { E } ( X ) = 3\) Fid \(\operatorname { Var } ( X )\).

4 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.

1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). The die is thrown 10 times.
2. Find the probability that there are not more than 4 throws on which the score is 1 .
3. Find the probability that there are exactly 4 throws on which the score is 2 .

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

3 The probability distribution of a random variable \(X\) is shown.

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Three independent values of \(X\), denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), are chosen. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 19\), write down all the possible sets of values for \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and hence find \(\mathrm { P } \left( X _ { 1 } = 7 \right)\).
3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5 s .

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)