IB DP Maths: AA HL复习笔记4.6.2 Calculations with Normal Distribution

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 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
• • 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?

• These are useful when:
• • The mean and/or standard deviation are unknown
  • You only have a diagram
  • You are working with the inverse 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'