5.02b Expectation and variance: discrete random variables

5 The discrete random variable \(U\) has probability distribution given by $$\mathrm { P } ( U = r ) = \begin{cases} \frac { 1 } { 16 } \binom { 4 } { r } & r = 0,1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find and simplify the probability generating function (pgf) of \(U\).
2. Use the pgf to find \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\).
3. Identify the distribution of \(U\), giving the values of any parameters.
4. Obtain the pgf of \(Y\), where \(Y = U ^ { 2 }\).
5. State, giving a reason, whether you can obtain the pgf of \(U + Y\) by multiplying the pgf of \(U\) by the pgf of \(Y\).

3 The discrete random variable \(X\) has probability generating function \(\frac { t } { a - b t }\), where \(a\) and \(b\) are constants.

1. Find a relationship between \(a\) and \(b\).
2. Use the probability generating function to find \(\mathrm { E } ( X )\) in terms of \(a\), giving your answer as simply as possible.
3. Expand the probability generating function as a power series, as far as the term in \(t ^ { 3 }\), giving the coefficients in terms of \(a\) and \(b\).
4. Name the distribution for which \(\frac { t } { a - b t }\) is the probability generating function, and state its parameter(s) in terms of \(a\).

2 The random variable \(X\) has the binomial distribution with parameters \(n\) and \(p\), i.e. \(X \sim \mathrm {~B} ( n , p )\).

1. Show that the probability generating function of \(X\) is \(\mathrm { G } ( t ) = ( q + p t ) ^ { n }\), where \(q = 1 - p\).
2. Hence obtain the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of \(X\).
3. Write down the mean and variance of the random variable \(Z = \frac { X - \mu } { \sigma }\).
4. Write down the moment generating function of \(X\) and use the linear transformation result to show that the moment generating function of \(Z\) is $$\mathrm { M } _ { Z } ( \theta ) = \left( q \mathrm { e } ^ { - \frac { p \theta } { \sqrt { n p q } } } + p \mathrm { e } ^ { \frac { q \theta } { \sqrt { n p q } } } \right) ^ { n } .$$
5. By expanding the exponential terms in \(\mathrm { M } _ { Z } ( \theta )\), show that the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(n \rightarrow \infty\) is \(\mathrm { e } ^ { \theta ^ { 2 } / 2 }\). You may use the result \(\lim _ { n \rightarrow \infty } \left( 1 + \frac { y + \mathrm { f } ( n ) } { n } \right) ^ { n } = \mathrm { e } ^ { y }\) provided \(\mathrm { f } ( n ) \rightarrow 0\) as \(n \rightarrow \infty\).
6. What does the result in part (v) imply about the distribution of \(Z\) as \(n \rightarrow \infty\) ? Explain your reasoning briefly.
7. What does the result in part (vi) imply about the distribution of \(X\) as \(n \rightarrow \infty\) ?

2 The random variable \(X ( X = 1,2,3,4,5,6 )\) denotes the score when a fair six-sided die is rolled.

1. Write down the mean of \(X\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\).
2. Show that \(\mathrm { G } ( t )\), the probability generating function (pgf) of \(X\), is given by $$\mathrm { G } ( t ) = \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) }$$ The random variable \(N ( N = 0,1,2 , \ldots )\) denotes the number of heads obtained when an unbiased coin is tossed repeatedly until a tail is first obtained.
3. Show that \(\mathrm { P } ( N = r ) = \left( \frac { 1 } { 2 } \right) ^ { r + 1 }\) for \(r = 0,1,2 , \ldots\).
4. Hence show that \(\mathrm { H } ( t )\), the pgf of \(N\), is given by \(\mathrm { H } ( t ) = ( 2 - t ) ^ { - 1 }\).
5. Use \(\mathrm { H } ( t )\) to find the mean and variance of \(N\). A game consists of tossing an unbiased coin repeatedly until a tail is first obtained and, each time a head is obtained in this sequence of tosses, rolling a fair six-sided die. The die is not rolled on the first occasion that a tail is obtained and the game ends at that point. The random variable \(Q ( Q = 0,1,2 , \ldots )\) denotes the total score on all the rolls of the die. Thus, in the notation above, \(Q = X _ { 1 } + X _ { 2 } + \ldots + X _ { N }\) where the \(X _ { i }\) are independent random variables each distributed as \(X\), with \(Q = 0\) if \(N = 0\). The pgf of \(Q\) is denoted by \(\mathrm { K } ( t )\). The familiar result that the pgf of a sum of independent random variables is the product of their pgfs does not apply to \(\mathrm { K } ( t )\) because \(N\) is a random variable and not a fixed number; you should instead use without proof the result that \(\mathrm { K } ( t ) = \mathrm { H } ( \mathrm { G } ( t ) )\).
6. Show that \(\mathrm { K } ( t ) = 6 \left( 12 - t - t ^ { 2 } - \ldots - t ^ { 6 } \right) ^ { - 1 }\).
   [0pt] [Hint. \(\left. \left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + \ldots + t ^ { 5 } \right) .\right]\)
7. Use \(\mathrm { K } ( t )\) to find the mean and variance of \(Q\).
8. Using your results from parts (i), (v) and (vii), verify the result that (in the usual notation for means and variances) $$\sigma _ { Q } { } ^ { 2 } = \sigma _ { N } { } ^ { 2 } \mu _ { X } { } ^ { 2 } + \mu _ { N } \sigma _ { X } { } ^ { 2 } .$$

1. The random variable \(X\) has probability function

$$\mathrm { P } ( X = x ) = \frac { x ^ { 2 } } { k } \quad x = 1,2,3,4$$

1. Show that \(k = 30\)
2. Find \(\mathrm { P } ( X \neq 4 )\)
3. Find the exact value of \(\mathrm { E } ( X )\)
4. Find the exact value of \(\operatorname { Var } ( X )\) Given that \(Y = 3 X - 1\)
5. find \(\mathrm { E } \left( Y ^ { 2 } \right)\)

1. The discrete random variable \(X\) has the probability distribution given in the table below.

1. Write down the value of \(\mathrm { F } ( 5 )\)
2. Find \(\mathrm { E } ( X )\)
3. Find \(\operatorname { Var } ( X )\) The random variable \(Y = 7 - 2 X\)
4. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
   3. \(\mathrm { P } ( Y > X )\) \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-03_2261_47_313_37}
      

1. In a game, the number of points scored by a player in the first round is given by the random variable \(X\) with probability distribution

Find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\)
3. \(\operatorname { Var } ( 3 - 2 X )\) The number of points scored by a player in the second round is given by the random variable \(Y\) and is independent of the number of points scored in the first round. The random variable \(Y\) has probability function $$\mathrm { P } ( Y = y ) = \frac { 1 } { 4 } \quad \text { for } y = 5,6,7,8$$
4. Write down the value of \(\mathrm { E } ( Y )\)
5. Find \(\mathrm { P } ( X = Y )\)
6. Find the probability that the number of points scored by a player in the first round is greater than the number of points scored by the player in the second round.

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

1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\) For this probability distribution, \(\operatorname { Var } ( X ) = \mathrm { E } \left( X ^ { 2 } \right)\)
2. 1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find the value of \(a\) and the value of \(b\)
3. Find \(\operatorname { Var } ( 1 - 3 X )\) Given that \(Y = 1 - X\), find
4. 1. \(\mathrm { P } ( Y < 0 )\)
   2. the largest possible value of \(k\) such that \(\mathrm { P } ( Y < k ) = 0.2\)

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

1. Find \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 4.54\)
2. find the value of \(a\) and the value of \(b\). The random variable \(Y = 3 - 2 X\)
3. Find \(\operatorname { Var } ( Y )\).

4. A spinner can land on the numbers \(10,12,14\) and 16 only and the probability of the spinner landing on each number is the same.
The random variable \(X\) represents the number that the spinner lands on when it is spun once.

1. State the name of the probability distribution of \(X\).
2. 1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find \(\operatorname { Var } ( X )\) A second spinner can land on the numbers 1, 2, 3, 4 and 5 only. The random variable \(Y\) represents the number that this spinner lands on when it is spun once. The probability distribution of \(Y\) is given in the table below

      \(y\)	1	2	3	4	5
      \(\mathrm { P } ( Y = y )\)	\(\frac { 4 } { 30 }\)	\(\frac { 9 } { 30 }\)	\(\frac { 6 } { 30 }\)	\(\frac { 5 } { 30 }\)	\(\frac { 6 } { 30 }\)

3. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\) The random variable \(W = a X + b\), where \(a\) and \(b\) are constants and \(a > 0\) Given that \(\mathrm { E } ( W ) = \mathrm { E } ( Y )\) and \(\operatorname { Var } ( W ) = \operatorname { Var } ( Y )\)
4. find the value of \(a\) and the value of \(b\). Each of the two spinners is spun once.
5. Find \(\mathrm { P } ( W = Y )\)

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

where \(a\) is a constant.

1. Find, in terms of \(a , \mathrm { E } ( X )\)
2. Find the range of the possible values of \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 0.56\)
3. find the possible values of \(a\)

1. The cumulative distribution of a discrete random variable \(X\) is given by

where \(k\) is a positive constant.

1. Show that \(k = 4\)
2. Find the probability distribution of the discrete random variable \(X\)
3. Using your answer to part (b), write down the mode of \(X\)
4. Calculate \(\operatorname { Var } ( 13 X - 6 )\)

5. The discrete random variable \(X\) has the following probability distribution where \(a\), \(b\) and \(c\) are probabilities.
Given that \(\mathrm { E } ( X ) = 0.8\)

1. find the value of \(c\). Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 5\) find
2. the value of \(a\) and the value of \(b\),
3. \(\operatorname { Var } ( X )\) The random variable \(Y = 5 - 3 X\) Find
4. \(\mathrm { E } ( Y )\)
5. \(\operatorname { Var } ( Y )\)
6. \(\mathrm { P } ( Y \geqslant 0 )\)

The random variable \(X\) has a discrete uniform distribution and takes the values \(1,2,3,4\) Find

1. \(\mathrm { F } ( 3 )\), where \(\mathrm { F } ( x )\) is the cumulative distribution function of \(X\),
2. \(\mathrm { E } ( X )\).
3. Show that \(\operatorname { Var } ( X ) = \frac { 5 } { 4 }\) The random variable \(Y\) has a discrete uniform distribution and takes the values $$3,3 + k , 3 + 2 k , 3 + 3 k$$ where \(k\) is a constant.
4. Write down \(\mathrm { P } ( Y = y )\) for \(y = 3,3 + k , 3 + 2 k , 3 + 3 k\) The relationship between \(X\) and \(Y\) may be written in the form \(Y = k X + c\) where \(c\) is a constant.
5. Find \(\operatorname { Var } ( Y )\) in terms of \(k\).
6. Express \(c\) in terms of \(k\).
   

3. A discrete random variable \(X\) has the probability function shown in the table below. Find

1. \(\mathrm { P } ( 1 < X \leq 3 )\),
2. \(\mathrm { F } ( 2.6 )\),
3. \(\mathrm { E } ( X )\),
4. \(\mathrm { E } ( 2 X - 3 )\),
5. \(\operatorname { Var } ( X )\)

8 The random variable \(S\) has the distribution \(\mathrm { B } ( 14 , p )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.3\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p > 0.3\). The critical region for the test is \(S \geqslant 8\).

1. Find the significance level of the test, correct to 3 significant figures.
2. It is given that, on each occasion that the test is carried out, the true value of \(p\) is equally likely to be \(0.3,0.5\) or 0.7 , independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.

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

5 The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} 0 & x < 0 \\ \frac{6}{5} x & 0 \leqslant x \leqslant 1 \\ \frac{6}{5} x^{-4} & x > 1 \end{cases}$$

1. Find \(\mathrm{P}(X > 1)\).
2. Find the median value of \(X\).
3. Given that \(\mathrm{E}(X) = 1\), find the variance of \(X\).
4. Find \(\mathrm{E}(\sqrt{X})\).

6 Aisha has a bag containing 3 red balls and 3 white balls. She selects a ball at random, notes its colour and returns it to the bag; the same process is repeated twice more. The number of red balls selected by Aisha is denoted by \(X\).

1. Find the probability generating function \(\mathrm{G}_{X}(t)\) of \(X\).

Basant also has a bag containing 3 red balls and 3 white balls. He selects three balls at random, without replacement, from his bag. The number of red balls selected by Basant is denoted by \(Y\).

1. Find the probability generating function \(\mathrm{G}_{Y}(t)\) of \(Y\).

The random variable \(Z\) is the total number of red balls selected by Aisha and Basant.

1. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
2. Use the probability generating function of \(Z\) to find \(\mathrm{E}(Z)\) and \(\operatorname{Var}(Z)\).

1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table. The spinner is spun twice. The total of the two numbers on which it lands is denoted by \(X\).

1. Show that \(\mathrm { P } ( X = 2 ) = 0.18\). The probability distribution of \(X\) is given in the table.

   \(x\)	0	1	2	3	4
   \(\mathrm { P } ( X = x )\)	0.49	0.28	0.18	0.04	0.01

2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

7 At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
2. Find
   1. \(\mathrm { P } ( X = 3 )\),
   2. \(\mathrm { P } ( X \geqslant 1 )\).
   3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .

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 \(\operatorname { Var } ( X ) = 4 a ( 1 - a )\).

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

1. Find \(p\).
2. Find \(\mathrm { E } ( X )\).

3

1. A random variable, \(X\), has the distribution \(\mathrm { B } ( 12,0.85 )\). Find
   1. \(\mathrm { P } ( X > 10 )\),
   2. \(\mathrm { P } ( X = 10 )\),
   3. \(\operatorname { Var } ( X )\).
   4. A random variable, \(Y\), has the distribution \(\mathrm { B } \left( 2 , \frac { 1 } { 4 } \right)\). Two independent values of \(Y\) are found. Find the probability that the sum of these two values is 1 .

5 A bag contains 4 blue discs and 6 red discs. Chloe takes a disc from the bag. If this disc is red, she takes 2 more discs. If not, she takes 1 more disc. Each disc is taken at random and no discs are replaced.

1. Complete the probability tree diagram in your Answer Book, showing all the probabilities. \includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-4_730_1203_529_511} The total number of blue discs that Chloe takes is denoted by \(X\).
2. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 5 }\). The complete probability distribution of \(X\) is given below.

   \(x\)	0	1	2
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 6 }\)	\(\frac { 3 } { 5 }\)	\(\frac { 7 } { 30 }\)

3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).