Calculate E(aX+b) or Var(aX+b) given distribution

Questions that provide a complete probability distribution (table or function) and ask to find E(aX+b) or Var(aX+b), requiring calculation of E(X) and/or Var(X) first.

3 questions · Moderate -0.8

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Edexcel S1 Specimen Q3
14 marks Moderate -0.8
The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} kx, & x = 1, 2, 3, 4, 5, \\ 0, & \text{otherwise.} \end{cases}$$
  1. Show that \(k = \frac{1}{15}\). [3]
Find the value of
  1. E\((2X + 3)\), [5]
  2. Var\((2X - 4)\). [6]
Edexcel S1 Q2
11 marks Moderate -0.8
The discrete random variable \(X\) has the probability function given by the following table:
\(x\)0123456
\(P(X = x)\)0.090.120.220.16\(p\)\(2p\)0.2
  1. Show that \(p = 0.07\) [2 marks]
  2. Find the value of \(E(X + 2)\). [4 marks]
  3. Find the value of \(\text{Var}(3X - 1)\). [5 marks]
OCR MEI Further Statistics Minor Specimen Q2
8 marks Moderate -0.8
The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.
\(x\)\(0\)\(1\)\(2\)\(3\)\(4\)
\(P(X = x)\)\(0.05\)\(0.2\)\(0.5\)\(0.2\)\(0.05\)
    1. Explain why \(\text{E}(X) = 2\). [1]
    2. Find \(\text{Var}(X)\). [3]
  2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250X - 80\). Find
The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.
  1. Explain why it would be inappropriate to test all the remote controls in this way. [1]
  2. State an advantage of using random sampling in this context. [1]