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.

4 questions

OCR S3 2009 January Q4
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
  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
  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
  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 \(V\)