State or write down basic properties

Questions asking to state, write down, or sketch the PDF, mean, median, or variance of a uniform distribution without requiring calculation or integration.

6 questions · Easy -1.1

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Edexcel S2 Q1
5 marks Easy -1.3
  1. (a) Explain the difference between a discrete and a continuous variable.
A random number generator on a calculator generates numbers, \(X\), to 3 decimal places, in the range 0 to 1 , e.g. 0.386 . The variable \(X\) may be modelled by a continuous uniform distribution, having the probability density function \(\mathrm { f } ( x )\), where $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1 \\ \mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$ (b) Explain why this model is not totally accurate.
(c) Sketch the cumulative distribution function of \(X\).
Edexcel S2 Q1
Easy -1.8
  1. It is estimated that \(4 \%\) of people have green eyes. In a random sample of size \(n\), the expected number of people with green eyes is 5 .
    1. Calculate the value of \(n\).
Edexcel S2 2009 January Q2
9 marks Easy -1.2
The continuous random variable \(X\) is uniformly distributed over the interval \([-2, 7]\).
  1. Write down fully the probability density function f(x) of \(X\). [2]
  2. Sketch the probability density function f(x) of \(X\). [2]
Find
  1. E(\(X^2\)), [3]
  2. P(\(-0.2 < X < 0.6\)). [2]
Edexcel S2 2004 June Q2
5 marks Easy -1.3
The continuous random variable \(X\) is uniformly distributed over the interval \([-1, 4]\). Find
  1. P\((X < 2.7)\), [1]
  2. E\((X)\), [2]
  3. Var \((X)\). [2]
Edexcel S2 Q7
19 marks Standard +0.3
Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.
  1. Find the mean and the standard deviation of \(X\). [3 marks]
It is suggested that a better model would be the distribution with probability density function $$f(x) = cx, \quad 0 \leq x \leq 5, \quad f(x) = c(10-x), \quad 5 < x \leq 10, \quad f(x) = 0 \text{ otherwise}.$$
  1. Write down the mean of \(X\). [1 mark]
  2. Find \(c\), and hence find the standard deviation of \(X\) in this model. [7 marks]
  3. Find P(\(4 < X < 6\)). [3 marks]
It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).
  1. Use these results to find P(\(4 < X < 6\)) in the third model. [4 marks]
  2. Compare your answer with (d). Which model do you think is most appropriate? [1 mark]
Edexcel S2 Q2
4 marks Easy -1.2
A searchlight is rotating in a horizontal circle. It is assumed that that, at any moment, the centre of its beam is equally likely to be pointing in any direction. The random variable \(X\) represents this direction, expressed as a bearing in the range \(000°\) to \(360°\).
  1. Specify a suitable model for the distribution of \(X\). [1 mark]
  2. Find the mean and the standard deviation of \(X\). [3 marks]