Calculate Var(aX+b) transformations

Questions that ask specifically for the variance of a linear transformation Var(aX+b) after finding or using Var(X), requiring the variance transformation formula.

5 questions

Edexcel S1 2017 January Q4
  1. In a game, the number of points scored by a player in the first round is given by the random variable \(X\) with probability distribution
\(x\)5678
\(\mathrm { P } ( X = x )\)0.130.210.290.37
Find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\)
  3. \(\operatorname { Var } ( 3 - 2 X )\) The number of points scored by a player in the second round is given by the random variable \(Y\) and is independent of the number of points scored in the first round. The random variable \(Y\) has probability function $$\mathrm { P } ( Y = y ) = \frac { 1 } { 4 } \quad \text { for } y = 5,6,7,8$$
  4. Write down the value of \(\mathrm { E } ( Y )\)
  5. Find \(\mathrm { P } ( X = Y )\)
  6. Find the probability that the number of points scored by a player in the first round is greater than the number of points scored by the player in the second round.
AQA Further AS Paper 2 Statistics 2021 June Q1
1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 6.5\)
Find \(\operatorname { Var } ( 4 X - 2 )\) Circle your answer.
2426102104
OCR Further Statistics AS 2018 June Q2
2 The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(\mathrm { P } ( W = w )\)0.250.36\(x\)\(x ^ { 2 }\)
  1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
  2. Find \(\operatorname { Var } ( 3 W + 6 )\).
OCR Further Statistics 2021 November Q2
2 A discrete random variable \(D\) has the following probability distribution, where \(a\) is a constant.
\(d\)0246
\(\mathrm { P } ( D = d )\)\(a\)0.10.30.2
Determine the value of \(\operatorname { Var } ( 3 D + 4 )\).
AQA Further AS Paper 2 Statistics 2020 June Q5
5 The discrete random variable \(X\) has the following probability distribution.
\(\boldsymbol { x }\)\(\mathbf { 2 }\)\(\mathbf { 4 }\)\(\mathbf { 6 }\)\(\mathbf { 9 }\)
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.20.60.10.1
5
  1. Find \(\mathrm { P } ( X \leq 6 )\) 5
  2. Let \(Y = 3 X + 2\)
    Show that \(\operatorname { Var } ( Y ) = 32.49\)
    5
  3. The continuous random variable \(T\) is independent of \(Y\). Given that \(\operatorname { Var } ( T ) = 5\), find \(\operatorname { Var } ( T + Y )\)