Uniform Distribution

3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$

1. Write down the name given to this distribution. Find
2. \(\mathrm { P } ( X = 4 )\)
3. \(\mathrm { F } ( 3 )\)
4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
5. Write down \(\mathrm { E } ( X )\)
6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
8. find \(\operatorname { Var } ( a X - 3 )\)

1 The discrete uniform distribution \(X\) can take values \(1,2,3 , \ldots , 10\)
Find \(\mathrm { P } ( X \geq 7 )\) Circle your answer. \(0.3 \quad 0.4 \quad 0.6 \quad 0.7\)

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

$$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } , \quad x = 1,2,3,4,5$$

1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 2\). Find
2. \(\mathrm { E } ( 3 X - 2 )\),
3. \(\operatorname { Var } ( 4 - 3 X )\).

5. The random variable \(X\) has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that \(\mathrm { E } ( X ) = 5\),

1. show that \(n = 9\). Find
2. \(\mathrm { P } ( X < 7 )\),
3. \(\operatorname { Var } ( X )\).

5 A fair spinner has five faces, labelled 0, 1, 2, 3, 4.

1. State the distribution of the score when the spinner is spun once.
2. Determine the probability that, when the spinner is spun twice, one of the scores is less than 2 and the other is at least 2.
3. Find the variance of the total score when the spinner is spun 5 times.

6. A discrete random variable is such that each of its values is assumed to be equally likely.

1. Write down the name of the distribution that could be used to model this random variable.
2. Give an example of such a distribution.
3. Comment on the assumption that each value is equally likely.
4. Suggest how you might refine the model in part (a).

4 The discrete random variable \(X\) has the distribution \(\mathrm { U } ( n )\).

1. Use the results \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) and \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\) to show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\). It is given that \(\mathrm { E } ( X ) = 13\).
2. Find the value of \(n\).
3. Find \(\mathrm { P } ( X < 7.5 )\). It is given that \(\mathrm { E } ( a X + b ) = 10\) and \(\operatorname { Var } ( a X + b ) = 117\), where \(a\) and \(b\) are positive.
4. Calculate the value of \(a\) and the value of \(b\).

4 A pack of \(k\) cards is labelled \(1,2 , \ldots , k\). A card is drawn at random from the pack. The random variable \(X\) represents the number on the card.

1. Given that \(k > 10\), find \(\mathrm { P } ( X \geqslant 10 )\). You are now given that \(k = 20\).
2. A card is drawn at random from the pack and the number on it is noted. The card is then returned to the pack. This process is repeated until the second occasion on which the number noted is less than 9 . Find the probability that no more than 4 cards have to be drawn. Answer all the questions. Section B (95 marks)

1 The random variable \(T\) follows a discrete uniform distribution and can take values \(1,2,3 , \ldots , 16\) Find the variance of \(T\) Circle your answer.
1.2518 .7521 .2521 .33

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

$$\mathrm { P } ( X = x ) = \frac { 1 } { \alpha } \quad \text { for } x = 1,2 , \ldots , \alpha$$ The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)

1. Show that \(2 \bar { X }\) is a biased estimator of \(\alpha\) A random sample of 6 observations of \(X\) is taken and the results are given below. $$\begin{array} { l l l l l l } 8 & 7 & 3 & 7 & 2 & 9 \end{array}$$
2. Use the sample mean to estimate the value of \(\alpha\)
   

4 A random number generator generates integers between 1 and 50 inclusive, with each number having an equal probability of being generated.

1. State the probability distribution of the numbers generated.
2. Determine the probability that a number generated is within one standard deviation of the mean.

5. The random variable \(X\) represents the number on the uppermost face when a fair die is thrown.

1. Write down the name of the probability distribution of \(X\).
2. Calculate the mean and the variance of \(X\). Three fair dice are thrown and the numbers on the uppermost faces are recorded.
3. Find the probability that all three numbers are 6 .
4. Write down all the different ways of scoring a total of 16 when the three numbers are added together.
5. Find the probability of scoring a total of 16 .

3. A fair six-sided die is rolled. The random variable \(Y\) represents the score on the uppermost, face.

1. Write down the probability function of \(Y\).
2. State the name of the distribution of \(Y\). Find the value of
3. \(\mathrm { E } ( 6 Y + 2 )\),
4. \(\operatorname { Var } ( 4 Y - 2 )\).

1. 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.
2. 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.
3. Find \(\operatorname { Var } ( Y )\) in terms of \(k\).
4. Express \(c\) in terms of \(k\).

6 The random variable \(X\) has a uniform distribution over the values \(\{ 1,4,7 , \ldots , 3 n - 2 \}\), where \(n\) is a positive integer.

1. Determine \(\operatorname { Var } ( X )\) in terms of \(n\).
2. Given that \(n = 100\), find the probability that \(X\) is within one standard deviation of the mean.