E(g(X)) and Var(g(X)) by integration

Questions requiring calculation of E(Y) or Var(Y) where Y = g(X) using the formula E(g(X)) = ∫g(x)f(x)dx with algebraic integration of the transformed function.

5 questions · Standard +0.4

5.03c Calculate mean/variance: by integration
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OCR S3 2009 January Q4
7 marks Standard +0.3
4 The weekly sales of petrol, \(X\) thousand litres, at a garage may be modelled by a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} c & 25 \leqslant x \leqslant 45 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant. The weekly profit, in \(\pounds\), is given by \(( 400 \sqrt { X } - 240 )\).
  1. Obtain the value of \(c\).
  2. Find the expected weekly profit.
  3. Find the probability that the weekly profit exceeds \(\pounds 2000\).
Edexcel S2 2016 June Q3
6 marks Moderate -0.8
  1. The random variable \(R\) has a continuous uniform distribution over the interval [5,9]
    1. Specify fully the probability density function of \(R\).
    2. Find \(\mathrm { P } ( 7 < R < 10 )\)
    The random variable \(A\) is the area of a circle radius \(R \mathrm {~cm}\).
  2. Find \(\mathrm { E } ( \mathrm { A } )\)
Edexcel FS2 2022 June Q7
7 marks Challenging +1.2
  1. A rectangle is to have an area of \(40 \mathrm {~cm} ^ { 2 }\)
The length of the rectangle, \(L \mathrm {~cm}\), follows a continuous uniform distribution over the interval [4, 10] Find the expected value of the perimeter of the rectangle.
Use algebraic integration, rather than your calculator, to evaluate any definite integrals.
Edexcel FS2 2023 June Q7
9 marks Standard +0.3
  1. The random variable \(R\) has a continuous uniform distribution over the interval \([ 2,10 ]\)
    1. Write down the probability density function \(\mathrm { f } ( r )\) of \(R\)
    A sphere of radius \(R \mathrm {~cm}\) is formed.
    The surface area of the sphere, \(S \mathrm {~cm} ^ { 2 }\), is given by \(S = 4 \pi R ^ { 2 }\)
  2. Show that \(\mathrm { E } ( S ) = \frac { 496 \pi } { 3 }\) The volume of the sphere, \(V \mathrm {~cm} ^ { 3 }\), is given by \(V = \frac { 4 } { 3 } \pi R ^ { 3 }\)
  3. Find, using algebraic integration, the expected value of \(V\)
OCR Further Pure Core 2 2018 March Q5
5 marks Standard +0.8
In this question you must show detailed reasoning. An ant starts from a fixed point \(O\) and walks in a straight line for \(1.5\) s. Its velocity, \(v\) cms\(^{-1}\), can be modelled by \(v = \frac{1}{\sqrt{9-t^2}}\). By finding the mean value of \(v\) in \(0 \leq t \leq 1.5\), deduce the average velocity of the ant. [5]