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IB DP Maths: AA HL复习笔记4.6.2 Calculations with Normal Distribution
Category:
IB课程
,
教材笔记
,
福利干货
Date: 2022年7月13日 上午11:51
Calculating Normal Probabilities
How do I find probabilities using a normal distribution?
How do I calculate
P(
X
=
x
): the probability of a single value for a normal distribution?
The probability of a
single value
is
always zero
for a normal distribution
You can picture this as the area of a single line is zero
Your G
DC
is likely to have a "
Normal Probability Density
" function
This is sometimes shortened to NPD, Normal PD or Normal Pdf
IGNORE THIS FUNCTION
for this course!
This calculates the
probability
density function
at a point
NOT the probability
How do I calculate
P(
a
<
X
<
b
): the probability of a range of values for a normal distribution?
You need a
GDC
that can calculate
cumulative normal probabilities
You want to use the "
Normal Cumulative Distribution
" function
This is sometimes shortened to NCD, Normal CD or Normal Cdf
You will need to enter:
The 'lower bound' - this is the value
a
The 'upper bound' - this is the value
b
The '
μ
' value - this is the mean
The '
σ
' value - this is the standard deviation
Check the order carefully
as some calculators ask for standard deviation before mean
Remember it is the standard deviation
so if you have the
variance
then
square root it
Always sketch
a quick diagram to visualise which area you are looking for
How do I calculate
P(
X
>
a
) or P(
X
<
b
) for a normal distribution?
You will still use the "
Normal Cumulative Distribution"
function
Using a value that is more than 4 standard deviations
smaller than the mean
is quite accurate
Or an easier option is just to input lots of 9's for the lower bound with a negative sign (
-99999999...
or
-10
99
)
Are there any useful identities?
These are useful when:
The mean and/or standard deviation are unknown
You only have a diagram
You are working with the
inverse distribution
Exam Tip
Check carefully whether you have entered the standard deviation or variance into your GDC
Worked Example
Inverse Normal Distribution
Given the value of
P(
X
<
a
) how do I find the value of
a
?
Your
GDC
will have a function called "
Inverse Normal Distribution
"
Some calculators call this InvN
The 'area' - this is the value
p
Some calculators might ask for the 'tail' - this is the left tail as you know the area to the left of
a
The '
μ
' value - this is the mean
The '
σ
' value - this is the standard deviation
Given the value of
P(
X
>
a
) how do I find the value of
a
?
Then use the
method for P(
X
<
a
)
to find
a
If your calculator does have the
tail option
(left, right or centre) then you can use the "Inverse Normal Distribution" function straightaway by:
Selecting 'right' for the tail
Entering the area as
'p'
Exam Tip
Always check your
answer makes sense
If P(
X
<
a
) is
less than 0.5
then
a
should be
smaller than the mean
If P(
X
<
a
) is
more than 0.5
then
a
should be
bigger than the mean
A sketch will help you see this
Worked Example
转载自savemyexams
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