5.02e Discrete uniform distribution

The continuous random variable \(T\) is equally likely to take any value from 5.0 to 11.0 inclusive.

1. Sketch the graph of the probability density function of \(T\). [2]
2. Write down the value of E(\(T\)) and find by integration the value of Var(\(T\)). [5]
3. A random sample of 48 observations of \(T\) is obtained. Find the approximate probability that the mean of the sample is greater than 8.3, and explain why the answer is an approximation. [6]

The random variable \(X\) follows a continuous uniform distribution over the interval \([2, 11]\).

1. Write down the mean of \(X\). [1 mark]
2. Find P(\(X \geq 8.6\)). [2 marks]
3. Find P(\(|X - 5| < 2\)). [2 marks]

The random variable \(Y\) follows a continuous uniform distribution over the interval \([a, b]\).

1. Show by integration that $$\text{E}(Y^2) = \frac{1}{3}(b^2 + ab + a^2).$$ [5 marks]
2. Hence, prove that $$\text{Var}(Y) = \frac{1}{12}(b - a)^2.$$ You may assume that E(\(Y\)) = \(\frac{1}{2}(a + b)\). [4 marks]

A class of children are each asked to draw a line that they think is 10 cm long without using a ruler. The teacher models how many centimetres each child's line is longer than 10 cm by the random variable \(X\) and believes that \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}, & -4 \leq x \leq 4, \\ 0, & \text{otherwise}. \end{cases}$$

1. Write down the name of this distribution. [1]
2. Define fully the cumulative distribution function F(x) of \(X\). [4]
3. Calculate the proportion of children making an error of less than 15\% according to this model. [3]
4. Give two reasons why this may not be a very suitable model. [2]

The random variable \(X\) represents the value on the upper face of an eight-sided dice after it has been rolled. The faces are numbered 1 to 8 The random variable \(X\) is modelled by a discrete uniform distribution with \(n = 8\)

1. Find E\((X)\) [1 mark]
2. Find Var\((X)\) [1 mark]
3. Find P\((X \geq 6)\) [1 mark]
4. The dice was rolled 800 times and the results below were obtained.

   \(x\)	1	2	3	4	5	6	7	8
   Frequency	103	63	84	110	74	41	85	240

   State, with a reason, how you would refine the model for the random variable \(X\). [2 marks]

The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that \(\text{E}(3X) = 30\) and \(\text{Var}(3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]

A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between \(0\) and \(10\) (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.

1. Let \(X\) be the number of points gained after shopping once. Find
2. Let \(Y\) be the number of points gained after shopping twice. Find
3. Find the probability of the most likely number of points gained after shopping twice. Justify your answer. [4]

The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that E\((3X) = 30\) and Var\((3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]