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.

7 questions

CAIE S2 2022 June Q6
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
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
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 S2 2009 January Q2
  1. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 2,7 ]\).
    1. Write down fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
    2. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\).
    Find
  2. \(\mathrm { E } \left( X ^ { 2 } \right)\),
  3. \(\mathrm { P } ( - 0.2 < X < 0.6 )\).
Edexcel S2 2004 June Q1
  1. Explain briefly what you understand by
    1. a sampling frame,
    2. a statistic.
    3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,4 ]\).
    Find
  2. \(\mathrm { P } ( X < 2.7 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\operatorname { Var } ( X )\).
Edexcel FS2 AS Specimen Q4
  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 \(\mathrm { E } \left( \mathrm { X } ^ { 3 } \right) = 17\)
OCR S2 2010 January Q7
7 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. Write down the value of \(\mathrm { E } ( T )\) and find by integration the value of \(\operatorname { Var } ( T )\).
  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.