5.02a Discrete probability distributions: general

A spinner is designed so that the score \(S\) is given by the following probability distribution.

1. Find the value of \(p\). [2]
2. Find \(\text{E}(S)\). [2]
3. Show that \(\text{E}(S^2) = 9.45\) [2]
4. Find \(\text{Var}(S)\). [2]

Tom and Jess play a game with this spinner. The spinner is spun repeatedly and \(S\) counters are awarded on the outcome of each spin. If \(S\) is even then Tom receives the counters and if \(S\) is odd then Jess receives them. The first player to collect 10 or more counters is the winner.

1. Find the probability that Jess wins after 2 spins. [2]
2. Find the probability that Tom wins after exactly 3 spins. [4]
3. Find the probability that Jess wins after exactly 3 spins. [3]

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

1. Given that \(E(X) = -0.2\), find the value of \(\alpha\) and the value of \(\beta\). [6]
2. Write down \(F(0.8)\). [1]
3. Evaluate \(\text{Var}(X)\). [4]

Find the value of

1. \(E(3X - 2)\), [2]
2. \(\text{Var}(2X + 6)\). [2]

The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} kx, & x = 1, 2, 3, 4, 5, \\ 0, & \text{otherwise.} \end{cases}$$

1. Show that \(k = \frac{1}{15}\). [3]

Find the value of

1. E\((2X + 3)\), [5]
2. Var\((2X - 4)\). [6]

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]

For a six-sided die it is assumed that each of the sides has an equal chance of landing uppermost when the die is rolled.

1. Write down the probability function for the random variable \(X\), the number showing on the uppermost side after the die has been rolled. [2]
2. State the name of the distribution. [1]

A student wishing to check the above assumption rolled the die 300 times and for the sides 1 to 6, obtained the frequencies 41, 49, 52, 58, 37 and 63 respectively.

1. Analyse these data and comment on whether or not the assumption is valid for this die. Use a 5\% level of significance and state your hypotheses clearly. [8]

The discrete random variable \(X\) can take any value in the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that P\((X = 1) = 0.125\) and that E\((X) = 4.5\). Beatrice finds that P\((X = 2) =\) P\((X = 3) =\) P\((X = 4) = p\). Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).

1. Calculate the values of \(p\) and \(q\). [7 marks]
2. Give the name for the distribution of \(X\). [1 mark]
3. Calculate the standard deviation of \(X\). [3 marks]

The discrete random variable \(X\) has the probability function given by the following table:

1. Show that \(p = 0.07\) [2 marks]
2. Find the value of \(E(X + 2)\). [4 marks]
3. Find the value of \(\text{Var}(3X - 1)\). [5 marks]

The discrete random variable \(X\) has probability function P\((X = x) = k(x + 4)\). Given that \(X\) can take any of the values \(-3, -2, -1, 0, 1, 2, 3, 4\),

1. find the value of the constant \(k\). [3 marks]
2. Find P\((X < 0)\). [2 marks]
3. Show that the cumulative distribution F\((x)\) is given by $$\text{F}(x) = \lambda(x + 4)(x + 5)$$ where \(\lambda\) is a constant to be found. [6 marks]

The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} cx^2 & x = -3, -2, -1, 1, 2, 3 \\ 0 & \text{otherwise.} \end{cases}$$

1. Show that \(c = \frac{1}{28}\). [3 marks]
2. Calculate
   1. \(E(X)\),
   2. \(E(X^2)\).
   [3 marks]
3. Calculate
   1. \(\text{Var}(X)\),
   2. \(\text{Var}(10 - 2X)\).
   [3 marks]

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

\(x\)	0	1	2	3	4	5
\(\text{P}(X = x)\)	0.11	0.17	0.2	0.13	\(p\)	\(p^2\)

1. Find the value of \(p\). [4 marks]
2. Find
   1. \(\text{P}(0 < X \leq 2)\),
   2. \(\text{P}(X \geq 3)\).
   [3 marks]
3. Find the mean and the variance of \(X\). [3 marks]
4. Construct a table to represent the cumulative distribution function \(\text{F}(x)\). [2 marks]

The random variable \(X\), which can take any value from \(\{1, 2, \ldots, n\}\), is modelled by the discrete uniform distribution with mean 10.

1. Show that \(n = 19\) and find the variance of \(X\). [4 marks]
2. Find \(\text{P}(3 < X \leq 6)\). [2 marks]

The random variable \(Y\) is defined by \(Y = 3(X - 10)\).

1. State the mean and the variance of \(Y\). [3 marks]

The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\text{f}(x) = \begin{cases} k(x + 1) & x = 1, 2, \ldots, 19, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{1}{209}\). [3 marks]
2. Find \(\text{P}(3 < X \leq 6)\) using this model. [3 marks]

A pack of 52 cards contains 4 cards bearing each of the integers from 1 to 13. A card is selected at random. The random variable \(X\) represents the number on the card.

1. Find \(P(X \leq 5)\). [1 mark]
2. Name the distribution of \(X\) and find the expectation and variance of \(X\). [4 marks]

A hand of 12 cards consists of three 2s, four 3s, two 4s, two 5s and one 6. The random variable \(Y\) represents the number on a card chosen at random from this hand.

1. Draw up a table to show the probability distribution of \(Y\). [3 marks]
2. Calculate \(\text{Var}(3Y - 2)\). [6 marks]

The discrete random variable \(X\) takes only the values \(4, 5, 6, 7, 8\) and \(9\). The probabilities of these values are given in the table:

\(x\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)
P\((X = x)\)	\(p\)	\(0.1\)	\(q\)	\(q\)	\(0.3\)	\(0.2\)

It is known that E\((X) = 6.7\). Find

1. the values of \(p\) and \(q\), [7 marks]
2. the value of \(a\) for which E\((2X + a) = 0\), [3 marks]
3. Var\((X)\). [3 marks]

A regular tetrahedron has its faces numbered 1, 2, 3 and 4. It is weighted so that when it is thrown, the probability of each face being in contact with the table is inversely proportional to the number on that face. This number is represented by the random variable \(X\).

1. Show that \(P(X = 1) = \frac{12}{25}\) and find the probabilities of the other values of \(X\). [5 marks]
2. Calculate the mean and the variance of \(X\). [5 marks]

A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.

\(x\)	0	1	2	3
P\((X = x)\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(\frac{1}{8}\)	\(\frac{1}{8}\)

1. Calculate E\((X)\) and Var\((X)\). [5]

The die is thrown 10 times.

1. Find the probability that there are not more than 4 throws on which the score is 1. [2]
2. Find the probability that there are exactly 4 throws on which the score is 2. [3]

When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values 2, 4, 6 and 8. The spinner is biased so that P(\(X = x\)) = \(kx\), where \(k\) is a constant.

1. Show that P(\(X = 6\)) = \(\frac{3}{10}\). [2]
2. Find E(\(X\)) and Var(\(X\)). [5]

A game at a charity event uses a bag containing 19 white counters and 1 red counter. To play the game once a player takes counters at random from the bag, one at a time, without replacement. If the red counter is taken, the player wins a prize and the game ends. If not, the game ends when 3 white counters have been taken. Niko plays the game once.

1. 1. Copy and complete the tree diagram showing the probabilities for Niko. [4] \includegraphics{figure_2}
   2. Find the probability that Niko will win a prize. [3]
2. The number of counters that Niko takes is denoted by \(X\).
   1. Find P(\(X = 3\)). [2]
   2. Find E(\(X\)). [4]

Each of four cards has a number printed on it as shown. Two of the cards are chosen at random, without replacement. The random variable \(X\) denotes the sum of the numbers on these two cards.

1. Show that P\((X = 6) = \frac{1}{6}\) and P\((X = 4) = \frac{1}{3}\). [3]
2. Write down all the possible values of \(X\) and find the probability distribution of \(X\). [4]
3. Find E\((X)\) and Var\((X)\). [5]

The probability distribution of a random variable \(X\) is shown.

\(x\)	1	3	5	7
P\((X = x)\)	0.4	0.3	0.2	0.1

1. Find E\((X)\) and Var\((X)\). [5]
2. Three independent values of \(X\), denoted by \(X_1\), \(X_2\) and \(X_3\), are chosen. Given that \(X_1 + X_2 + X_3 = 19\), write down all the possible sets of values for \(X_1\), \(X_2\) and \(X_3\) and hence find P\((X_1 = 7)\). [2]
3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5s. [3]

In her purse, Katharine has two £5 notes, two £10 notes and one £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 \(£X\).

1. 1. Show that P(X = 10) = 0.1. [1]
   2. Show that P(X = 30) = 0.2. [2]
   The table shows the probability distribution of X.

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

2. Find E(X) and Var(X). [5]

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

1. Show that \(k = \frac{1}{70}\). [2]
2. Find E\((X)\) and Var\((X)\). [5]