Linear transformation to achieve target parameters

1 Exam marks, \(X\), have mean 70 and standard deviation 8.7. The marks need to be scaled using the formula \(Y = a X + b\) so that the scaled marks, \(Y\), have mean 55 and standard deviation 6.96. Find the values of \(a\) and \(b\).

7

1. Random variables \(Y\) and \(X\) are related by \(Y = a + b X\), where \(a\) and \(b\) are constants and \(b > 0\). The standard deviation of \(Y\) is twice the standard deviation of \(X\). The mean of \(Y\) is 7.92 and is 0.8 more than the mean of \(X\). Find the values of \(a\) and \(b\).
2. Random variables \(R\) and \(S\) are such that \(R \sim \mathrm {~N} \left( \mu , 2 ^ { 2 } \right)\) and \(S \sim \mathrm {~N} \left( 2 \mu , 3 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( R + S > 1 ) = 0.9\).
   1. Find \(\mu\).
   2. Hence find \(\mathrm { P } ( S > R )\).

1 The marks obtained by a randomly chosen student in the two papers of an examination are denoted by the random variables \(X\) and \(Y\), where \(X \sim \mathrm {~N} ( 45,81 )\) and \(Y \sim \mathrm {~N} ( 33,63 )\). The student's overall mark for the examination, \(T\), is given by \(T = X + \lambda Y\), where the constant \(\lambda\) is chosen such that \(\mathrm { E } ( T ) = 100\).

1. Show that \(\lambda = \frac { 5 } { 3 }\).
2. Assuming that \(X\) and \(Y\) are independent, state the distribution of \(T\), giving the values of its parameters.
3. Comment on the assumption of independence.

5 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)\) and \(\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)\) respectively. The random variable \(a X + b Y\), where \(a\) and \(b\) are constants, has the distribution \(\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)\).

1. Given that \(a\) and \(b\) are integers, find the value of \(a\) and the value of \(b\).
2. Given that \(\mathrm { P } ( X > Y ) = 0.966\), find \(\sigma ^ { 2 }\).

8 The Normal variable \(X\) is transformed to the Normal variable \(Y\).
The transformation is \(\mathrm { y } = \mathrm { a } + \mathrm { bx }\), where \(a\) and \(b\) are positive constants.
You are given that \(X \sim N ( 42,6.8 )\) and \(Y \sim N ( 57.2,11.492 )\).
Determine the values of \(a\) and \(b\).

1. The random variable \(X \sim \mathrm {~N} \left( 5,0.4 ^ { 2 } \right)\) and the random variable \(Y \sim \mathrm {~N} \left( 8,0.1 ^ { 2 } \right)\) \(X\) and \(Y\) are independent random variables.
   A random sample of \(a\) independent observations is taken from the distribution of \(X\) and one observation is taken from the distribution of \(Y\)

The random variable \(W = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { a } + b Y\) and has the distribution \(\mathrm { N } \left( 169,2 ^ { 2 } \right)\) Find the value of \(a\) and the value of \(b\)

\(X\) is a continuous random variable with the distribution N\((48.5, 12.5^2)\). The values of \(X\) are transformed to standardised values of \(Y\), using the equation \(Y = aX + b\), where \(a\) and \(b\) are constants with \(a > 0\).

1. Find values of \(a\) and \(b\) for which the mean and standard deviation of \(Y\) are 40 and 10 respectively. [4]
2. State the distribution of \(Y\). [1]

Two randomly chosen standardised values are denoted by \(Y_1\) and \(Y_2\).

1. Calculate the probability that \(Y_2\) is at least 10 greater than \(Y_1\). [5]