Throughout this section we will use the random variable . For a normal distribution,?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 and ?is equal to the?probability
  • Remember for a normal distribution ?so you do not need to worry about whether the inequality is strict (< or >) or weak (≤ or ≥)
• The equation of a normal distribution curve is complicated so the?area must be calculated numerically
• You will be?expected to use?distribution functions on your?calculator?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
• P(X = x?) = 0
• Your calculator 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?calculator?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
• P(X > a) 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)
• Similarly P(X < b) 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)
• This works because the probability that?X?is more than 3 standard deviations bigger than the mean is less than 0.0015
  • This is the same for being 3 standard deviations less than the mean
  • This reduces to less than 0.000032 when using 4 standard deviations

Are there any useful identities?

• 
• As ?you can use:
  • 
  • 
  • 
• 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 calculator will have a function called "Inverse Normal Distribution"
  • Some calculators call this InvN
• Given that P(X < a) =?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
• 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

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

• Given P(X > a) =?p
• Use P(X < a) = 1 - P(X > a)??to rewrite this as P(X < a) = 1 -?p
• Then use the method for?(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'