Variance of linear transformation

A question is this type if and only if it asks to find Var(aX + b) where X follows a discrete uniform distribution and a, b are constants.

4 questions

OCR Further Statistics AS 2024 June Q1
1 The random variable \(W\) can take values 1,2 or 3 and has a discrete uniform distribution.
  1. Write down the value of \(\mathrm { E } ( 2 W )\).
  2. Find the value of \(\operatorname { Var } ( 2 W )\).
  3. Determine the value of the constant \(k\) for which \(\mathrm { E } ( 2 \mathrm {~W} + \mathrm { k } ) = \operatorname { Var } ( 2 \mathrm {~W} + \mathrm { k } )\). The random variable \(S\) has the probability distribution shown in the following table.
    \(S\)23456
    \(P ( S = S )\)\(\frac { 2 } { 9 }\)\(\frac { 1 } { 9 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 9 }\)\(\frac { 2 } { 9 }\)
  4. Calculate \(\operatorname { Var } ( S )\).
Edexcel S1 2006 June Q4
  1. The random variable \(X\) has the discrete uniform distribution
$$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } , \quad x = 1,2,3,4,5$$
  1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 2\). Find
  2. \(\mathrm { E } ( 3 X - 2 )\),
  3. \(\operatorname { Var } ( 4 - 3 X )\).
OCR MEI Further Statistics A AS 2020 November Q4
4 A fair 8 -sided dice has faces labelled \(1,2 , \ldots , 8\). The random variable \(X\) represents the score when the dice is rolled once.
  1. State the distribution of \(X\).
  2. Find \(\mathrm { P } ( X < 4 )\).
  3. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    • The random variable \(Y\) is defined by \(Y = 10 X + 5\). Find each of the following.
    • \(\mathrm { E } ( Y )\)
    • \(\operatorname { Var } ( Y )\)
AQA Further Paper 3 Statistics 2020 June Q4
4 The discrete random variable \(X\) follows a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\). The discrete random variable \(Y\) is defined by \(Y = 2 X\)
4
  1. Using the standard results for \(\sum n , \sum n ^ { 2 }\) and \(\operatorname { Var } ( a X + b )\), prove that $$\operatorname { Var } ( Y ) = \frac { n ^ { 2 } - 1 } { 3 }$$ 4
  2. A spinning toy can land on one of four values: 2, 4, 6 or 8
    Using a discrete uniform distribution, find the probability that the next value the toy lands on is greater than 2 4
  3. State an assumption that is required for the discrete uniform distribution used in part (b) to be valid.