IB DP Maths: AI HL复习笔记4.8.2 Calculations with Normal Distribution

• 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
• 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

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

Given the value of P(X > a)  how do I find the value of a?