5.02b Expectation and variance: discrete random variables

3 The discrete random variable \(R\) has the following probability distribution. It is known that \(\mathrm { E } ( R ) = 0.2\) and \(\operatorname { Var } ( R ) = 3.56\) Find the values of \(a , b\) and \(c\).
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2 The probability distribution for the discrete random variable \(W\) is given in the table.

1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
2. Find \(\operatorname { Var } ( 3 W + 6 )\).

1 When a spinner is spun, the outcome is equally likely to be 1,2 or 3 . In a competition, the spinner is spun twice and the outcomes are added to give a total score \(T\).

1. Show that the expectation of \(T\) is 4 .
2. Find the variance of \(T\). A competitor pays \(\pounds 1.50\) to enter the competition and receives \(\pounds X\), where \(X = 0.3 T\).
3. 1. Find the expectation of the competitor's profit.
   2. Find the variance of the competitor's profit.

3 A discrete random variable \(X\) has the following probability distribution.

1. Determine the value of \(p\).
2. It is given that \(\mathrm { E } ( a X + b ) = \operatorname { Var } ( a X + b ) = 23.19\), where \(a\) and \(b\) are positive constants. Determine the value of \(a\) and the value of \(b\).

4 A discrete random variable \(W\) has the probability distribution shown in the following table, in which \(a\) and \(b\) are constants. It is given that \(\mathrm { E } ( W - 60 ) = 0.15\). Determine the value of \(\operatorname { Var } ( 4 W - 60 )\).

1 The random variable \(W\) can take values 1,2 or 3 and has a discrete uniform distribution.

1. Write down the value of \(\mathrm { E } ( 2 W )\).
2. Find the value of \(\operatorname { Var } ( 2 W )\).
3. Determine the value of the constant \(k\) for which \(\mathrm { E } ( 2 \mathrm {~W} + \mathrm { k } ) = \operatorname { Var } ( 2 \mathrm {~W} + \mathrm { k } )\). The random variable \(S\) has the probability distribution shown in the following table.

   \(S\)	2	3	4	5	6
   \(P ( S = S )\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)

4. Calculate \(\operatorname { Var } ( S )\).

2 Every time a spinner is spun, the probability that it shows the number 4 is 0.2 , independently of all other spins.

1. A pupil spins the spinner repeatedly until it shows the number 4. Find the mean of the number of spins required.
2. Calculate the probability that the number of spins required is between 3 and 10 inclusive.
3. Each pupil in a class of 30 spins the spinner until it shows the number 4. Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\).

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.

1 A discrete random variable \(X\) has the following distribution, where \(a , b\) and \(c\) are constants. It is given that \(\mathrm { E } ( X ) = 1.25\) and \(\operatorname { Var } ( X ) = 0.8875\).

1. Determine the values of \(a\), \(b\) and \(c\).
2. The random variable \(Y\) is defined by \(Y = 7 - 2 X\). Write down the value of \(\operatorname { Var } ( Y )\).
3. Twenty independent observations of \(X\) are obtained. The number of those observations for which \(X = 3\) is denoted by \(T\). Find the value of \(\operatorname { Var } ( T )\).

2 A discrete random variable \(D\) has the following probability distribution, where \(a\) is a constant. Determine the value of \(\operatorname { Var } ( 3 D + 4 )\).

3 A game is played as follows. A fair six-sided dice is thrown once. If the score obtained is even, the amount of money, in \(\pounds\), that the contestant wins is half the score on the dice, otherwise it is twice the score on the dice.

1. Find the probability distribution of the amount of money won by the contestant.
2. The contestant pays \(\pounds 5\) for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice.

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

1. The discrete random variable \(X\) represents the score when a biased spinner is spun. The probability distribution of \(X\) is given by

where \(p\) and \(q\) are probabilities.

1. Find \(\mathrm { E } ( X )\). Given that \(\operatorname { Var } ( X ) = 2.5\)
2. find the value of \(p\).
3. Hence find the value of \(q\). Amar is invited to play a game with the spinner.
   The spinner is spun once and \(X _ { 1 }\) is the score on the spinner. If \(X _ { 1 } > 0\) Amar wins the game.
   If \(X _ { 1 } = 0\) Amar loses the game.
   If \(X _ { 1 } < 0\) the spinner is spun again and \(X _ { 2 }\) is the score on this second spin and if \(X _ { 1 } + X _ { 2 } > 0\) Amar wins the game, otherwise Amar loses the game.
4. Find the probability that Amar wins the game. Amar does not want to lose the game.
   He says that because \(\mathrm { E } ( X ) > 0\) he will play the game.
5. State, giving a reason, whether or not you would agree with Amar.

1. The discrete random variable \(X\) takes the values \(- 1,2,3,4\) and 7 only.

Given that $$\mathrm { P } ( X = x ) = \frac { 8 - x } { k } \text { for } x = - 1,2,3,4 \text { and } 7$$ find the value of \(\mathrm { E } ( X )\)

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

Given that \(\mathrm { E } ( X ) = 0.5\)

1. find the value of \(a\). Given also that \(\operatorname { Var } ( X ) = 5.01\)
2. find the value of \(b\) and the value of \(c\). The random variable \(Y = 5 - 8 X\)
3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
4. Find \(\mathrm { P } \left( 4 X ^ { 2 } > Y \right)\)

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.

2. A spinner can land on the numbers \(2,4,5,7\) or 8 only. The random variable \(X\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(X\) is given in the table below.

1. Find \(\mathrm { P } ( 2 X - 3 > 5 )\) Given that \(\mathrm { E } ( X ) = 4.6\)
2. show that \(\operatorname { Var } ( X ) = 4.14\) The random variable \(Y = a X - b\) where \(a\) and \(b\) are positive constants.
   Given that $$\mathrm { E } ( Y ) = 13.4 \quad \text { and } \quad \operatorname { Var } ( Y ) = 66.24$$
3. find the value of \(a\) and the value of \(b\) In a game Sam and Alex each spin the spinner once, landing on \(X _ { 1 }\) and \(X _ { 2 }\) respectively.
   Sam's score is given by the random variable \(S = X _ { 1 }\) Alex's score is given by the random variable \(R = 2 X _ { 2 } - 3\) The person with the higher score wins the game. If the scores are the same it is a draw.
4. Find the probability that Sam wins the game.

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

where \(a\) and \(b\) are constants.

1. Write down an equation for \(a\) and \(b\).
2. Calculate \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 3.5\)
3. 1. find a second equation in \(a\) and \(b\),
   2. hence find the value of \(a\) and the value of \(b\).
4. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 5 - 3 X\)
5. Find \(\mathrm { P } ( Y > 0 )\).
6. Find
   1. \(\mathrm { E } ( Y )\),
   2. \(\operatorname { Var } ( Y )\).

1. The discrete random variable \(X\) is defined by the cumulative distribution function

where \(k\) is a constant.

1. Find the probability distribution of \(X\).
2. Find \(\mathrm { P } ( 1.5 < X \leqslant 3.5 )\) The random variable \(Y = 12 - 7 X\)
3. Calculate Var(Y)
4. Calculate \(\mathrm { P } ( 4 X \leqslant | Y | )\)

1. Adana selects one number at random from the distribution of \(X\) which has the following probability distribution.

1. Given that the number selected by Adana is not 5 , write down the probability it is 0
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 75\)
3. Find \(\operatorname { Var } ( X )\)
4. Find \(\operatorname { Var } ( 4 - 3 X )\) Bruno and Charlie each independently select one number at random from the distribution of \(X\)
5. Find the probability that the number Bruno selects is greater than the number Charlie selects. Devika multiplies Bruno's number by Charlie's number to obtain a product, \(D\)
6. Determine the probability distribution of \(D\)

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

3. A fair six-sided die is rolled. The random variable \(Y\) represents the score on the uppermost, face.

1. Write down the probability function of \(Y\).
2. State the name of the distribution of \(Y\). Find the value of
3. \(\mathrm { E } ( 6 Y + 2 )\),
4. \(\operatorname { Var } ( 4 Y - 2 )\).

4. The random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = k x , \quad x = 1,2 , \ldots , 5$$

1. Show that \(k = \frac { 1 } { 15 }\). Find
2. \(\mathrm { P } ( X < 4 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } ( 3 X - 4 )\).