Edexcel S2 (Statistics 2) 2003 June

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable \(Y \sim \text{Po}(30)\).

1. Estimate P(\(Y > 28\)). [6]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of \(X\). [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(\(X\)) and Var (\(X\)). [3]

1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]

A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that

1. 1. the first 5 will occur on the sixth throw,
   2. in the first eight throws there will be exactly three 5s.
   [8]

A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml. The random variable \(X\) is the volume of lemonade dispensed into a cup.

1. Specify the probability density function of \(X\) and sketch its graph. [4]
2. Find the probability that the machine dispenses
   1. less than 183 ml,
   2. exactly 183 ml.
   [3]
3. Calculate the inter-quartile range of \(X\). [1]
4. Determine the value of \(x\) such that P(\(X \geq x\)) = 2P(\(X \leq x\)). [3]
5. Interpret in words your value of \(x\). [2]

A continuous random variable \(X\) has probability density function f(\(x\)) where $$\text{f}(x) = \begin{cases} k(x^2 + 2x + 1) & -1 \leq x \leq 0, \\ 0, & \text{otherwise} \end{cases}$$ where \(k\) is a positive integer.

1. Show that \(k = 3\). [4]

Find

1. E(\(X\)), [4]
2. the cumulative distribution function F(\(x\)), [4]
3. P(\(-0.3 < X < 0.3\)). [3]