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IB DP Maths: AI HL复习笔记4.6.1 Linear Combinations of Random Variables
Category:
IB课程
,
教材笔记
,
福利干货
Date: 2022年7月21日 上午11:02
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
) =
a
E(
X
) ± b
Var(
aX ±
b
) =
a
² 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
) =
a
E(
X
) +
b
E(
Y
)
This is true for
any random variables
X
and
Y
Var(
aX
+
bY
) =
a
² Var(
X
) +
b
² Var(
Y
)
This is true if
X
and
Y
are
independent
E(
aX
-
bY
) =
a
E(
X
) -
b
E(
Y
)
Var(
aX
-
bY
) =
a
² Var(
X
) +
b
² Var(
Y
)
Notice that you still add the two terms together on the right hand side
This is because
b
² is positive even if
b
is negative
Therefore the variances of
aX
+
bY
and
aX
-
bY
are the same
What is the mean and variance of a linear combination of
n
random variables?
Let
X
1
, X
2
, ..., X
n
be
n
random variables and
a
1
, a
2
, ..., a
n
be
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 2
X
and
X
1
+ X
2
?
2
X
means
one observation
of
X
is taken and
then doubled
X
1
+ X
2
means
two observations
of
X
are taken and then
added together
2
X
and
X
1
+ X
2
have the
same expected values
E(2
X
) = 2E(
X
)
E(
X
1
+ X
2
) = E(
X
1
) + E(
X
2
) = 2E(
X
)
2
X
and
X
1
+ X
2
have
different variances
Var(2
X
) = 2²Var(
X
) = 4Var(
X
)
Var(
X
1
+ X
2
) = Var(
X
1
) + Var(
X
2
) = 2Var(
X
)
To see the distinction:
Suppose
X
could take the values 0 and 1
2
X
could then take the values 0 and 2
X
1
+ X
2
could then take the values 0, 1 and 2
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
+
E
1
+
E
2
+
E
3
+
E
4
+
E
5
+
E
6
where
C
is the mass of a carton
E
is the mass of an egg
It is
not
C
+ 6
E
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:
X
1
+
X
2
+ ... +
X
n
You are taking a
single observation
of a variable and multiplying it by a constant:
n
X
Worked Example
转载自savemyexams
Previous post: IB DP Maths: AI HL复习笔记4.5.2 Expected Values
Next post: IB EE论文怎么写才能拿高分?
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