Linear combinations of independent variables

1 The mean and variance of the random variable \(X\) are 5.8 and 3.1 respectively. The random variable \(S\) is the sum of three independent values of \(X\). The independent random variable \(T\) is defined by \(T = 3 X + 2\).

1. Find the variance of \(S\).
2. Find the variance of \(T\).
3. Find the mean and variance of \(S - T\).

1 The random variable \(X\) has mean 2.4 and variance 3.1.

1. The random variable \(Y\) is the sum of four independent values of \(X\). Find the mean and variance of \(Y\).
2. The random variable \(Z\) is defined by \(Z = 4 X - 3\). Find the mean and variance of \(Z\).

3 The discrete random variable \(W\) has the distribution \(\mathrm { U } ( 11 )\). The independent discrete random variable \(V\) has the distribution \(\mathrm { U } ( 5 )\).

1. It is given that, for constants \(m\) and \(n\), with \(m > 0\), \(\mathrm { E } ( \mathrm { mW } + \mathrm { nV } ) = 0\) and \(\operatorname { Var } ( \mathrm { mW } + \mathrm { nV } ) = 1\). Determine the exact values of \(m\) and \(n\). The random variable \(T\) is the mean of three independent observations of \(W\).
2. Explain whether the Central Limit Theorem can be used to say that the distribution of \(T\) is approximately normal.

6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables: The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).

1. Tabulate the probability distribution for \(Z\).
2. Calculate \(\mathrm { E } ( Z )\).
3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
4. Calculate Var (3Z-4).

7 A fair coin has + 1 written on the heads side and - 1 on the tails side. The coin is tossed 100 times. The sum of the numbers showing on the 100 tosses is the random variable \(Y\). Show that the variance of \(Y\) is 100 . [4] \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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4 The table below shows the probability distribution for the number of students, \(R\), attending classes for a particular mathematics module.

1. Find values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
2. The number of students, \(S\), attending classes for a different mathematics module is such that $$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$ Find values for the mean and variance of:
   1. \(T = R + S\);
   2. \(\quad D = S - R\).