Calculate probabilities and expectations

Questions requiring calculation of probabilities, expectations of functions of X, or variance using integration or formulas for a given uniform distribution.

4 questions · Moderate -0.4

5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration
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CAIE S2 2022 June Q6
9 marks Moderate -0.3
6 A random variable \(X\) has probability density function f . The graph of \(\mathrm { f } ( x )\) is a straight line segment parallel to the \(x\)-axis from \(x = 0\) to \(x = a\), where \(a\) is a positive constant.
  1. State, in terms of \(a\), the median of \(X\).
  2. Find \(\mathrm { P } \left( X > \frac { 3 } { 4 } a \right)\).
  3. Show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }\).
  4. Given that \(\mathrm { P } ( X < b ) = p\), where \(0 < b < \frac { 1 } { 2 } a\), find \(\mathrm { P } \left( \frac { 1 } { 3 } b < X < a - \frac { 1 } { 3 } b \right)\) in terms of \(p\).
Edexcel S2 2017 June Q7
9 marks Moderate -0.3
7. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)
  1. Find an expression, in terms of \(a\) and \(b\), for \(\mathrm { E } ( 3 - 2 X )\)
  2. Find \(\mathrm { P } \left( X > \frac { 1 } { 3 } b + \frac { 2 } { 3 } a \right)\) Given that \(\mathrm { E } ( X ) = 0\)
  3. find an expression, in terms of \(b\) only, for \(\mathrm { E } \left( 3 X ^ { 2 } \right)\) Given also that the range of \(X\) is 18
  4. find \(\operatorname { Var } ( X )\)
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Edexcel S2 2006 January Q3
8 marks Easy -1.2
3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).
  1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
  2. \(\mathrm { E } ( X )\),
  3. \(\operatorname { Var } ( \mathrm { X } )\),
  4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).
Edexcel FS2 AS Specimen Q4
8 marks Standard +0.3
  1. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 3,5 ]\).
    1. Sketch the probability density function \(\mathrm { f } ( \mathrm { x } )\) of X .
    2. Find the value of k such that \(\mathrm { P } ( \mathrm { X } < 2 [ \mathrm { k } - \mathrm { X } ] ) = 0.25\)
    3. Use algebraic integration to show that \(\mathrm { E } \left( \mathrm { X } ^ { 3 } \right) = 17\)