Calculations with Normal Distribution (DP IB Maths: AI HL)

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Dan

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Dan

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Calculating Normal Probabilities

Throughout this section we will use the random variable X tilde straight N left parenthesis mu comma space sigma squared right parenthesis. For X distributed normally, X can take any real number. Therefore any values mentioned in this section will be assumed to be real numbers.

How do I find probabilities using a normal distribution?

  • The area under a normal curve between the points x equals a and x equals b is equal to the probability straight P left parenthesis a less than X less than b right parenthesis
    • Remember for a normal distribution you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)
      • straight P left parenthesis a less than X less than b right parenthesis equals straight P left parenthesis a less or equal than X less or equal than b right parenthesis
  • You will be expected to use distribution functions on your GDC to find the probabilities when working with 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
  • straight P left parenthesis X equals x right parenthesis equals 0
  • Your GDC 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
  • straight P left parenthesis X greater than a right parenthesis can be estimated using an upper bound that is sufficiently bigger than the mean
    • Using a value that is more than 4 standard deviations bigger than the mean is quite accurate
    • Or an easier option is just to input lots of 9's for the upper bound (99999999... or 1099)
  • straight P left parenthesis X less than b right parenthesis can be estimated using a lower bound that is sufficiently smaller than the mean
    • 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 -1099)

Are there any useful identities?

  • straight P left parenthesis X less than mu right parenthesis equals straight P left parenthesis X greater than mu right parenthesis equals 0.5
  • As straight P left parenthesis X equals a right parenthesis equals 0 you can use:
    • straight P left parenthesis X less than a right parenthesis plus straight P left parenthesis X greater than a right parenthesis equals 1
    • straight P left parenthesis X greater than a right parenthesis equals 1 minus straight P left parenthesis X less than a right parenthesis
    • straight P left parenthesis a less than X less than b right parenthesis equals straight P left parenthesis X less than b right parenthesis minus straight P left parenthesis X less than a right parenthesis
  • These are useful when:
    • The mean and/or standard deviation are unknown
    • You only have a diagram
    • You are working with the inverse distribution

Examiner Tip

  • Check carefully whether you have entered the standard deviation or variance into your GDC

Worked example

The random variable Y tilde straight N left parenthesis 20 comma 5 squared right parenthesis. Calculate:

i)
straight P left parenthesis Y equals 20 right parenthesis.

4-6-2-ib-ai-aa-sl-normal-prob-a-we-solution

ii)
straight P left parenthesis 18 less or equal than Y less than 27 right parenthesis.

4-6-2-ib-ai-aa-sl-normal-prob-b-we-solution

iii)
straight P left parenthesis Y greater than 29 right parenthesis

4-6-2-ib-ai-aa-sl-normal-prob-c-we-solution

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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
  • Given that straight P left parenthesis X less than a right parenthesis equals p you will need to enter:
    • 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? 

  • 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'
  • If your calculator does not have the tail option (left, right or centre) then:
    • Given straight P left parenthesis X greater than a right parenthesis equals p
    • Use straight P left parenthesis X less than a right parenthesis equals 1 minus straight P left parenthesis X greater than a right parenthesis to rewrite this as
      • straight P left parenthesis X less than a right parenthesis equals 1 minus p
    • Then use the method for P(X < a) to find a

Examiner 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

The random variable  W tilde straight N left parenthesis 50 comma space 36 right parenthesis.

Find the value of w such that straight P left parenthesis W greater than w right parenthesis equals 0.175.

4-6-2-ib-ai-aa-sl-inverse-normal-we-solution

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.