Discrete Random Variables

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

5. A bag contains a large number of counters with \(35 \%\) of the counters having a value of 6 and \(65 \%\) of the counters having a value of 9 A random sample of size 2 is taken from the bag and the value of each counter is recorded as \(X _ { 1 }\) and \(X _ { 2 }\) respectively. The statistic \(Y\) is calculated using the formula $$Y = \frac { 2 X _ { 1 } + X _ { 2 } } { 3 }$$

1. List all the possible values of \(Y\).
2. Find the sampling distribution of \(Y\).
3. Find \(\mathrm { E } ( Y )\).

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 \(Z\).

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

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

1. (a) Define a statistic.

A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { \mathrm { n } }\) is taken from a population with unknown mean \(\mu\).
(b) For each of the following state whether or not it is a statistic.

1. \(\frac { X _ { 1 } + X _ { 4 } } { 2 }\),
2. \(\frac { \sum X ^ { 2 } } { n } - \mu ^ { 2 }\).

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.

3. The discrete random variable \(X\) has the probability distribution given below.

1. Find the mean of \(X\) in terms of \(k\).
2. Find the bias in using ( \(2 \bar { X } - 5\) ) as an estimator of \(k\). Fifty observations of \(X\) were made giving a sample mean of 8.34 correct to 3 significant figures.
3. Calculate an unbiased estimate of \(k\).
   (2 marks)

6. 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 \(\frac { 2 } { 3 } X _ { 1 } - \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(\mu\). An unbiased estimator for \(\mu\) is given by \(\hat { \mu } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
2. Show that \(\operatorname { Var } ( \hat { \mu } ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }\).
3. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { \mu }\) has 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
2. (ii) Given that \(c = 0.1\), find the value of \(\mathrm { E } ( T )\)
   [0pt] [3 marks]
   6
3. Show that \(\mathrm { E } ( \sqrt { | T | } ) = \frac { 5 \sqrt { 2 } + 52 \sqrt { 3 } } { 50 }\)
   [0pt] [3 marks]

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 \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).

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 \(\mathrm { P } ( Y = 3 \mid X = 1 )\).

1. A bag contains a large number of \(10 \mathrm { p } , 20 \mathrm { p }\) and 50 p 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 Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of a continuous random variable with probability density function $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta
0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a parameter whose value is to be estimated.

1. Find \(\mathrm { E } ( X )\).
2. Show that \(S _ { 1 } = X _ { 1 } + X _ { 2 }\) is an unbiased estimator of \(\theta\).
   \(L\) is the larger of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal.
3. Show that the probability density function of \(L\) is \(\frac { 2 l } { \theta ^ { 2 } }\) for \(0 \leqslant l \leqslant \theta\).
4. Find \(\mathrm { E } ( L )\).
5. Find an unbiased estimator \(S _ { 2 }\) of \(\theta\), based on \(L\).
6. Determine which of the two estimators \(S _ { 1 }\) and \(S _ { 2 }\) is the more efficient.

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.

6 The continuous random variable \(X\) has mean \(\mu\) and variance \(\sigma ^ { 2 }\), and the independent continuous random variable \(Y\) has mean \(2 \mu\) and variance \(3 \sigma ^ { 2 }\). Two observations of \(X\) and three observations of \(Y\) are taken and are denoted by \(X _ { 1 } , X _ { 2 } , Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) respectively.

1. Find the expectation of the sum of these 5 observations and hence construct an unbiased estimator, \(T _ { 1 }\), of \(\mu\).
2. The estimator \(T _ { 2 }\), where \(T _ { 2 } = X _ { 1 } + X _ { 2 } + c \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)\), is an unbiased estimator of \(\mu\). Find the value of the constant \(c\).
3. Determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is more efficient.
4. Find the values of the constants \(a\) and \(b\) for which $$a \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } \right) + b \left( Y _ { 1 } ^ { 2 } + Y _ { 2 } ^ { 2 } + Y _ { 3 } ^ { 2 } \right)$$ is an unbiased estimator of \(\sigma ^ { 2 }\).

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.
   • \(\mathrm { P } ( X - 2 Y > 0 )\)
   • \(\mathrm { P } ( X - 2 Y > 1 )\)
   • How could the estimates in part (a) be improved?
   The mean of 50 values of \(X - 2 Y\) is denoted by the random variable \(W\).
2. Calculate an estimate of \(\mathrm { P } ( W > 1 )\).