Compare uniform with other distributions

Questions asking to compare properties of uniform distribution with normal or other distributions, or to assess model suitability.

4 questions · Moderate -0.1

5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf
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OCR S2 2007 June Q7
10 marks Moderate -0.3
7 Two continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
  1. Sketch on the same axes the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). [You should not use graph paper or attempt to plot points exactly.]
  2. Explain in everyday terms the difference between the two random variables.
  3. Find the value of \(t\) such that \(\mathrm { P } ( T > t ) = 0.2\).
OCR S2 2009 January Q5
9 marks Standard +0.3
5 The continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \\ T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
  1. Sketch, on the same axes, the graphs of f and g .
  2. Describe in everyday terms the difference between the distributions of the random variables \(S\) and \(T\). (Answers that comment only on the shapes of the graphs will receive no credit.)
  3. Calculate the variance of \(T\).
Edexcel Paper 3 2018 June Q1
5 marks Moderate -0.8
  1. Helen believes that the random variable \(C\), representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.
    1. Write down the probability distribution for \(C\).
    2. Using this model, find the probability that cloud cover is less than 50\%
    Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than \(50 \%\) was 0.315
  2. Comment on the suitability of Helen's model in the light of this information.
  3. Suggest an appropriate refinement to Helen's model.
Edexcel S2 Q5
13 marks Standard +0.3
The random variable \(X\) has a continuous uniform distribution on the interval \(a \leq X \leq 3a\).
  1. Without assuming any standard results, prove that \(\mu\), the mean value of \(X\), is equal to \(2a\) and derive an expression for \(\sigma^2\), the variance of \(X\), in terms of \(a\). [7 marks]
  2. Find the probability that \(|X - \mu| < \sigma\) and compare this with the same probability when \(x\) is modelled by a Normal distribution with the same mean and variance. [6 marks]