IB DP Maths: AI HL復習筆記4.6.1 Linear Combinations of Random Variables
Transformation of a Single Variable
What is Var(X)?
- Var(X) represents the variance of the random variable?X
- Var(X) can be calculated by the formula
You will?not be required?to use this formula in the exam
What are the formulae for E(aX?±?b) and Var(aX?±?b)?
- If?a?and?b?are constants then the following formulae are true:
- E(aX?±?b) =?aE(X) ± b
- Var(aX ±?b) =?a2 Var(X)
- These are given in the?formula booklet
- This is the same as linear transformations of data
- The mean is affected by multiplication and addition/subtraction
- The variance is affected by multiplication but not addition/subtraction
- Remember division can be written as a multiplication
Worked Example
Transformation of Multiple Variables
What is the mean and variance of?aX?+?bY?
- Let?X?and?Y?be two random variables and let?a?and?b?be two constants
- E(aX?+?bY) =?aE(X) +?bE(Y)
- This is true for?any random variables?X?and?Y
- Var(aX?+?bY) =?a2 Var(X) +?b2 Var(Y)
- This is true if?X?and?Y?are?independent
- E(aX?-?bY) =?aE(X) -?bE(Y)
- Var(aX?-?bY) =?a2 Var(X) +?b2 Var(Y)
- Notice that you still add the two terms together on the right hand side
- This is because?b2 is positive even if?b?is negative
- Therefore the variances of?aX?+?bY?and?aX?-?bY?are the same
- Notice that you still add the two terms together on the right hand side
What is the mean and variance of a linear combination of?n?random variables?
- Let?X1, X2, ..., Xn?be?n?random variables and?a1, a2, ..., anbe?n?constants
- This is true if the random variables are?independent
- Notice that the constants get squared so the terms on the right-hand side will always be positive
For a given random variable?X,?what is the difference between 2X?and?X1?+ X2?
- 2X?means?one observation?of?X?is taken and?then doubled
- X1?+ X2means?two observations?of?X?are taken and then?added together
- 2X?and?X1?+ X2?have the?same expected values
- E(2X) = 2E(X)
- E(X1?+ X2) = E(X1) + E(X2) = 2E(X)
- 2X?and?X1?+ X2?have?different variances
- Var(2X) = 22Var(X) = 4Var(X)
- Var(X1?+ X2) = Var(X1) + Var(X2) = 2Var(X)
- To see the distinction:
- Suppose?X?could take the values 0 and 1
- 2X?could then take the values 0 and 2
- X1?+ X2?could then take the values 0, 1 and 2
- Suppose?X?could take the values 0 and 1
- Questions are likely to describe the variables in content
- For example: The mass of a carton containing 6 eggs is the mass of the carton plus the mass of the 6?individual?eggs
- This can be modelled by?M?=?C?+?E1+?E2?+?E3?+?E4?+?E5?+?E6?where
- C?is the mass of a carton
- E?is the mass of an egg
- It is?not?C?+ 6E?because the masses of the 6 eggs could be?different
Exam Tip
- In an exam when dealing with multiple variables ask yourself which of the two cases is true
- You are adding together?different observations?using the same variable:?X1?+?X2?+ ... +?Xn
- You are taking a?single observation?of a variable and multiplying it by a constant:?nX
Worked Example
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