Distribution of linear combination

2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\). Distribution of \(A\) : \(\_\_\_\_\)
Distribution of \(B\) : \(\_\_\_\_\)

7. The random variable \(X\) is defined as $$X = 4 Y - 3 W$$ where \(Y \sim \mathrm {~N} \left( 40,3 ^ { 2 } \right) , W \sim \mathrm {~N} \left( 50,2 ^ { 2 } \right)\) and \(Y\) and \(W\) are independent.

1. Find \(\mathrm { P } ( X > 25 )\) The random variables \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) are independent and each has the same distribution as \(Y\). The random variable \(A\) is defined as $$A = \sum _ { i = 1 } ^ { 3 } Y _ { i }$$ The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 115 , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( A - C < 0 ) = 0.2\) and that \(A\) and \(C\) are independent,
2. find the variance of \(C\).

6. The random variable \(W\) is defined as $$W = 3 X - 4 Y$$ where \(X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Given that \(\mathrm { P } ( W < 44 ) = 0.9\)

1. find the value of \(\sigma\), giving your answer to 2 decimal places. The random variables \(A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) each have the same distribution as \(A\), where \(A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)\) The random variable \(B\) is defined as $$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$ where \(X , A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) are independent.
2. Find \(\mathrm { P } ( B \leqslant 145 \mid B > 120 )\)

7. The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} \left( 30,4.5 ^ { 2 } \right) \text { and } Y \sim \mathrm {~N} \left( 20,3.5 ^ { 2 } \right)$$ The random variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent and each has the same distribution as \(X\). The random variables \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent and each has the same distribution as \(Y\). Given that the random variable \(A\) is defined as $$A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 5 }$$

1. find \(\mathrm { P } ( A < 24 )\) The random variable \(W\) is such that \(W \sim \mathrm {~N} \left( \mu , 2.8 ^ { 2 } \right)\) Given that \(\mathrm { P } ( W - X < 4 ) = 0.1\) and that \(W\) and \(X\) are independent,
2. find the value of \(\mu\), giving your answer to 3 significant figures.

1. The random variable \(X\) is defined as

$$X = 4 A - 3 B$$ where \(A\) and \(B\) are independent and $$A \sim \mathrm {~N} \left( 15,5 ^ { 2 } \right) \quad B \sim \mathrm {~N} \left( 10,4 ^ { 2 } \right)$$

1. Find \(\mathrm { P } ( X < 40 )\) The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 20 , \sigma ^ { 2 } \right)\)
   The random variables \(C _ { 1 } , C _ { 2 }\) and \(C _ { 3 }\) are independent and each has the same distribution as \(C\) The random variable \(D\) is defined as $$D = \sum _ { i = 1 } ^ { 3 } C _ { i }$$ Given that \(\mathrm { P } ( A + B + D < 76 ) = 0.2420\) and that \(A , B\) and \(D\) are independent,
2. showing your working clearly, find the standard deviation of \(C\)

1. The weights of a particular type of apple, \(A\) grams, and a particular type of orange, \(R\) grams, each follow independent normal distributions.

$$A \sim \mathrm {~N} \left( 160,12 ^ { 2 } \right) \quad R \sim \mathrm {~N} \left( 140,10 ^ { 2 } \right)$$

1. Find the distribution of
   1. \(A + R\)
   2. the total weight of 2 randomly selected apples. A box contains 4 apples and 1 orange only. Jesse selects 2 pieces of fruit at random from the box.
2. Find the probability that the total weight of the 2 pieces of fruit exceeds 310 grams. From a large number of apples and oranges, Celeste selects \(m\) apples and 1 orange at random. The random variable \(W\) is given by $$W = \left( \sum _ { i = 1 } ^ { m } A _ { i } \right) - n \times R$$ where \(n\) is a positive integer.
   Given that the middle \(95 \%\) of the distribution of \(W\) lies between 1100.08 and 1499.92 grams, (c) find the value of \(m\) and the value of \(n\)