5.02a Discrete probability distributions: general

5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.

1. Find the probability generating function of \(X\).
   Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
   

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3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).

2 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 5 } + p t + q t ^ { 2 }$$ where \(p\) and \(q\) are constants.

1. Given that \(\mathrm { E } ( X ) = 1.1\), find the numerical value of \(\operatorname { Var } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-04_2714_38_109_2010} The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t )\) given by $$\mathrm { G } _ { Y } ( t ) = \frac { 2 } { 3 } t \left( 1 + \frac { 1 } { 2 } t ^ { 2 } \right)$$ The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
2. Find the probability generating function of \(Z\).
3. Find \(\mathrm { P } ( Z > 2 )\).
4. State the most probable value of \(Z\).

5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.

1. Find the probability generating function of \(X\).
   Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
   

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3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).

4 The table below shows the probability distribution of the random variable \(X\).

1. Find the value of the constant \(k\).
2. Calculate the values of \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 Part of the probability distribution of a variable, \(X\), is given in the table.

1. Find \(\mathrm { P } ( X = 0 )\).
2. Find \(\mathrm { E } ( X )\).

4 Each of the variables \(W , X , Y\) and \(Z\) takes eight integer values only. The probability distributions are illustrated in the following diagrams. \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_437_394_397_280} \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_433_380_397_685} \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_428_383_402_1082} \includegraphics[max width=\textwidth, alt={}, center]{43f7e091-9ae7-4373-a209-e2ebdba5260f-3_425_376_402_1482}

1. For which one or more of these variables is
   1. the mean equal to the median,
   2. the mean greater than the median?
   3. Give a reason why none of these diagrams could represent a geometric distribution.
   4. Which one of these diagrams could not represent a binomial distribution? Explain your answer briefly.

5 The probability distribution of a discrete random variable, \(X\), is given in the table. It is given that the expectation, \(\mathrm { E } ( X )\), is \(1 \frac { 1 } { 4 }\).

1. Calculate the values of \(p\) and \(q\).
2. Calculate the standard deviation of \(X\).

1 The table shows the probability distribution for a random variable X. Calculate \(\mathrm { E } ( \mathrm { X } )\) and \(\operatorname { Var } ( \mathrm { X } )\).

1 The table shows the probability distribution of a random variable \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. Three values of \(X\) are chosen at random. Find the probability that \(X\) takes the value 2 at least twice.

5 A sixth-form class consists of 7 girls and 5 boys. Three students from the class are chosen at random. The number of boys chosen is denoted by the random variable \(X\). Show that

1. \(\quad \mathrm { P } ( X = 0 ) = \frac { 7 } { 44 }\),
2. \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 22 }\). The complete probability distribution of \(X\) is shown in the following table.

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)	\(\frac { 7 } { 44 }\)	\(\frac { 21 } { 44 }\)	\(\frac { 7 } { 22 }\)	\(\frac { 1 } { 22 }\)

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

4 The number, \(X\), of children per family in a certain city is modelled by the probability distribution \(\mathrm { P } ( X = r ) = k ( 6 - r ) ( 1 + r )\) for \(r = 0,1,2,3,4\).

1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac { 1 } { 50 }\).

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(6 k\)	\(10 k\)

2. Calculate \(\mathrm { E } ( X )\).
3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children.

2 Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.

1. Explain why the case \(X = 3\) is impossible.
2. Explain why \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 24 }\).
3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } - 1 \right) \text { for } r = 2,3,4,5 .$$

1. Show the probability distribution in a table, and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 Yasmin has 5 coins. One of these coins is biased with P (heads) \(= 0.6\). The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, \(X\).

1. Show that \(\mathrm { P } ( X = 0 ) = 0.025\).
2. Show that \(\mathrm { P } ( X = 1 ) = 0.1375\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)	0.025	0.1375	0.3	0.325	0.175	0.0375

3. Draw a vertical line chart to illustrate the probability distribution.
4. Comment on the skewness of the distribution.
5. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
6. Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 .

2 A couple plan to have at least one child of each sex, after which they will have no more children. However, if they have four children of one sex, they will have no more children. You should assume that each child is equally likely to be of either sex, and that the sexes of the children are independent. The random variable \(X\) represents the total number of girls the couple have.

1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 11 } { 16 }\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 16 }\)	\(\frac { 11 } { 16 }\)	\(\frac { 1 } { 8 }\)	\(\frac { 1 } { 16 }\)	\(\frac { 1 } { 16 }\)

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

4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( r + 1 ) \quad \text { for } r = 1,2,3,4,5 .$$

1. Show that \(k = \frac { 1 } { 70 }\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4$$

1. Show that \(k = 0.05\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

1 In her purse, Katharine has two \(\pounds 5\) notes, two \(\pounds 10\) notes and one \(\pounds 20\) note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(\pounds X\).

1. (A) Show that \(\mathrm { P } ( X = 10 ) = 0.1\).
   (B) Show that \(\mathrm { P } ( X = 30 ) = 0.2\). The table shows the probability distribution of \(X\).

   \(r\)	10	15	20	25	30
   \(\mathrm { P } ( X = r )\)	0.1	0.4	0.1	0.2	0.2

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

3 The table shows the probability distribution of the random variable \(X\).

1. Explain why \(\mathrm { E } ( X ) = 25\).
2. Calculate \(\operatorname { Var } ( X )\).

4 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.

1. Find the value of \(p\).
2. Find the expectation and variance of \(X\).
3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.

7 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable \(X\) represents the number of sites investigated. The probability distribution of \(X\) is shown below.

1. Find the expectation and variance of \(X\).
2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
3. Draw a suitable diagram to illustrate the distribution of \(X\).

3 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6$$

1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(k\)					\(11 k\)

2. Find the mean of \(X\). A fair six-sided die is rolled three times.
3. Find the probability that the total score is 16 .

5 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 The discrete random variable \(X\) takes the values 0 and 1 with \(\mathrm { P } ( X = 0 ) = q\) and \(\mathrm { P } ( X = 1 ) = p\), where \(p + q = 1\).

1. Write down the probability generating function of \(X\). The sum of \(n\) independent observations of \(X\) is denoted by \(S\).
2. Write down the probability generating function of \(S\), and name the distribution of \(S\).
3. Use the probability generating function of \(S\) to find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
4. The independent random variables \(Y\) and \(Z\) are such that \(Y\) has the distribution \(\mathrm { B } \left( 10 , \frac { 1 } { 2 } \right)\), and \(Z\) has probability generating function \(\mathrm { e } ^ { - ( 1 - t ) }\). Find the probability that the sum of one random observation of \(Y\) and one random observation of \(Z\) is equal to 2 .