Discrete Random Variables

A fair coin has \(+1\) written on the heads side and \(-1\) on the tails side. The coin is tossed \(100\) times. The sum of the numbers showing on the \(100\) tosses is the random variable \(Y\). Show that the variance of \(Y\) is \(100\). [4]

1 The random variable \(X\) has mean 2.4 and variance 3.1.

1. The random variable \(Y\) is the sum of four independent values of \(X\). Find the mean and variance of \(Y\).
2. The random variable \(Z\) is defined by \(Z = 4 X - 3\). Find the mean and variance of \(Z\).

The random variable \(X\) has the binomial distribution B(12, 0·3). The independent random variable \(Y\) has the Poisson distribution Po(4). Find

1. \(E(XY)\), [2]
2. Var\((XY)\). [6]

1. The probability distribution of the discrete random variable \(X\) is

$$P ( X = x ) = \begin{cases} \frac { k } { x } & \text { for } x = 1,2 \text { and } 3 \\ \frac { m } { 2 x } & \text { for } x = 6 \text { and } 9 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(m\) are positive constants.
Given that \(\mathrm { E } ( X ) = 3.8\), find \(\operatorname { Var } ( X )\)

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

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

1. Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\) Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
3. find the value of \(q\) and the value of \(r\)
4. Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)

1. A fair six-sided black die has faces numbered \(1,2,2,3,3\) and 4

The random variable \(B\) represents the score when the black die is rolled.

1. Write down the value of \(\mathrm { E } ( B )\) A white die has 6 faces numbered \(1,1,2,4,5\) and \(c\) where \(c > 5\) The discrete random variable \(W\) represents the score when the white die is rolled and has probability distribution given by

   \(w\)	1	2	4	5	\(c\)
   \(\mathrm { P } ( W = w )\)	\(a + b\)	\(a\)	0.3	\(a\)	\(b\)

   Greg and Nilaya play a game with these dice.
   Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
   The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
2. Find the value of \(a\) and the value of \(b\) Show your working clearly. The random variable \(X = 2 W - 5\) Given that \(\mathrm { E } ( X ) = 2.6\)
3. find the exact value of \(\operatorname { Var } ( X )\)

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1. A bag contains a large number of counters.

40\% of the counters are numbered 1 \(35 \%\) of the counters are numbered 2 \(25 \%\) of the counters are numbered 3 In a game Alif draws two counters at random from the bag. His score is 4 times the number on the first counter minus 2 times the number on the second counter.

1. Show that Alif gets a score of 8 with probability 0.0875
2. Find the sampling distribution of Alif's score.
3. Calculate Alif's expected score.

Explain briefly what you understand by

1. a statistic, [2]
2. a sampling distribution. [2]

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

1. Find \(\operatorname { Var } ( X )\)
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\)

The continuous random variable \(X\) is uniformly distributed over the interval \((\theta - 1, \theta + 5)\), where \(\theta\) is an unknown constant.

1. Find the mean and the variance of \(X\). [2]
2. Let \(\overline{X}\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\overline{X}\), an unbiased estimator for \(\theta\) and determine its standard error. [4]

A random sample of three independent variables \(X_1, X_2\) and \(X_3\) is taken from a distribution with mean \(\mu\) and variance \(\sigma^2\).

1. Show that \(\frac{1}{3}X_1 + \frac{1}{3}X_2 + \frac{1}{3}X_3\) is an unbiased estimator for \(\mu\). [3]

An unbiased estimator for \(\mu\) is given by \(\hat{\mu} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.

1. [(b)] Show that Var(\(\hat{\mu}\)) = \((2a^2 - 2a + 1)\sigma^2\). [6]
2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{\mu}\) has minimum variance. [5]

6. A bag contains a large number of balls. 65\% are numbered 1 35\% are numbered 2 A random sample of 3 balls is taken from the bag.
Find the sampling distribution for the range of the numbers on the 3 selected balls.

A random sample \(X_1, X_2, ..., X_{10}\) is taken from a population with mean \(\mu\) and variance \(\sigma^2\).

1. Determine the bias, if any, of each of the following estimators of \(\mu\). $$\theta_1 = \frac{X_1 + X_4 + X_5}{3}$$ $$\theta_2 = \frac{X_{10} - X_1}{3}$$ $$\theta_3 = \frac{3X_1 + 2X_5 + X_{10}}{6}$$ [4]
2. Find the variance of each of these estimators. [5]
3. State, giving reasons, which of these three estimators for \(\mu\) is
   1. the best estimator,
   2. the worst estimator.
   [4]

6. A continuous uniform distribution on the interval \([ 0 , k ]\) has mean \(\frac { k } { 2 }\) and variance \(\frac { k ^ { 2 } } { 12 }\). A random sample of three independent variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is taken from this distribution.

1. Show that \(\frac { 2 } { 3 } X _ { 1 } + \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(k\). An unbiased estimator for \(k\) is given by \(\hat { k } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
2. Show that \(\operatorname { Var } ( \hat { k } ) = \left( a ^ { 2 } - 2 a + 2 \right) \frac { k ^ { 2 } } { 6 }\)
3. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { k }\) has minimum variance, and calculate this minimum variance.

1 An unbiased coin is tossed three times. The random variables \(F\) and \(S\) denote the total number of heads that occur in the first two tosses and the total number of heads that occur in the last two tosses respectively. The table above shows the joint probability distribution of \(F\) and \(S\).

1. Show how the entry \(\frac { 1 } { 4 }\) in the table is obtained.
2. Find \(\operatorname { Cov } ( F , S )\).

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

1. Find \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) is defined in terms of the discrete random variable \(X\) When \(X\) is negative, \(Y = X ^ { 2 }\) When \(X\) is positive, \(Y = 3 X - 2\)
2. Find \(\mathrm { P } ( Y < 9 )\)
3. Find \(\mathrm { E } ( X Y )\)

6 The random variable \(T\) can take the value \(T = - 2\) or any value in the range \(0 \leq T < 12\) The distribution of \(T\) is given by \(\mathrm { P } ( T = - 2 ) = c , \mathrm { P } ( 0 \leq T \leq t ) = 225 k - k ( 15 - t ) ^ { 2 }\) 6

1. 1. Show that \(1 - c = 216 k\) [0pt] [3 marks] 6
      1. (ii) Given that \(c = 0.1\), find the value of \(\mathrm { E } ( T )\) [0pt] [3 marks]
         6
   2. Show that \(\mathrm { E } ( \sqrt { | T | } ) = \frac { 5 \sqrt { 2 } + 52 \sqrt { 3 } } { 50 }\) [0pt] [3 marks]

5 Two discrete random variables \(X\) and \(Y\) have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where \(a\) is a constant.

1. Show that \(a = \frac { 1 } { 27 }\).
2. Find \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Cov } ( X , Y )\).
4. Are \(X\) and \(Y\) independent? Give a reason for your answer.
5. Find \(\mathrm { P } ( X = 1 \mid Y = 2 )\).

8 The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) have the distributions \(\mathrm { B } \left( n _ { 1 } , \theta \right)\) and \(\mathrm { B } \left( n _ { 2 } , \theta \right)\) respectively. Two possible estimators for \(\theta\) are $$T _ { 1 } = \frac { 1 } { 2 } \left( \frac { X _ { 1 } } { n _ { 1 } } + \frac { X _ { 2 } } { n _ { 2 } } \right) \text { and } T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { n _ { 1 } + n _ { 2 } } .$$

1. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are both unbiased estimators, and calculate their variances.
2. Find \(\frac { \operatorname { Var } \left( T _ { 1 } \right) } { \operatorname { Var } \left( T _ { 2 } \right) }\). Given that \(n _ { 1 } \neq n _ { 2 }\), use the inequality \(\left( n _ { 1 } - n _ { 2 } \right) ^ { 2 } > 0\) to find which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.

5 The independent discrete random variables \(U\) and \(V\) can each take the values 1, 2 and 3, all with probability \(\frac { 1 } { 3 }\). The random variables \(X\) and \(Y\) are defined as follows: $$X = | U - V | , Y = U + V .$$

1. In the Printed Answer Book complete the table showing the joint probability distribution of \(X\) and \(Y\).
2. Find \(\operatorname { Cov } ( X , Y )\).
3. State with a reason whether \(X\) and \(Y\) are independent.
4. Find \(\mathrm { P } ( Y = 3 \mid X = 1 )\).

9 The continuous random variable \(X\) has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2 \\ 1 & x > 2 \end{array} \right.$$ The discrete random variable \(Y\) has the probability distribution shown below. The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).

A bag contains a large number of 10p, 20p and 50p coins in the ratio 1 : 2 : 2 A random sample of 3 coins is taken from the bag. Find the sampling distribution of the median of these samples. [7]

6 The random variables \(S\) and \(T\) are independent and have joint probability distribution given in the table.

1. Show that \(a = 0.12\) and find the value of \(b\).
2. Find \(\mathrm { P } ( T - S = 1 )\).
3. Find \(\operatorname { Var } ( T - S )\).

1 For the variables \(A\) and \(B\), it is given that \(\operatorname { Var } ( A ) = 9 , \operatorname { Var } ( B ) = 6\) and \(\operatorname { Var } ( 2 A - 3 B ) = 18\).

1. Find \(\operatorname { Cov } ( A , B )\).
2. State with a reason whether \(A\) and \(B\) are independent.

2 Boxes of matches contain 50 matches. Full boxes have mean mass 20.0 grams and standard deviation 0.4 grams. Empty boxes have mean mass 12.5 grams and standard deviation 0.2 grams. Stating any assumptions that you need to make, calculate the mean and standard deviation of the mass of a match. [7]

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.

10 The discrete random variables \(X\) and \(Y\) have distributions as follows: \(X \sim \mathrm {~B} ( 20,0.3 )\) and \(Y \sim \operatorname { Po } ( 3 )\). The spreadsheet in Fig. 10 shows a simulation of the distributions of \(X\) and \(Y\). Each of the 20 rows below the heading row consists of a value of \(X\), a value of \(Y\), and the value of \(X - 2 Y\). \begin{table}[h] \end{table}

1. Use the spreadsheet to estimate each of the following.
   The mean of 50 values of \(X - 2 Y\) is denoted by the random variable \(W\).
2. Calculate an estimate of \(\mathrm { P } ( W > 1 )\).