CIE A Level Maths: Probability & Statistics 2复习笔记2.4.2 Approximations of Distributions

When can I use a normal distribution to approximate a Poisson distribution?

• The greater the value of λ in a Poisson distribution, the more symmetrical the distribution becomes and the closer it resembles the bell-shaped curve of a normal distribution

What are continuity corrections?

• The Poisson distribution is discrete and the normal distribution is continuous
• A continuity correction takes this into account when using a normal approximation
• The probability being found will need to be changed from a discrete variable, X to a continuous variable, X_N

Do I need to use continuity corrections?

• Yes!How do I approximate a probability?

• STEP 1: Find the mean and variance of the approximating distributionSTEP 2: Apply continuity corrections to the inequality
• STEP 3: Find the probability of the new corrected inequality
  • Find the standard normal probability and use the table of the normal distribution
• The probability will not be exact as it is an approximation but provided λ is large enough (approximately > 15) then it will be a close approximation

When can I use a Poisson distribution to approximate a binomial distribution?

• A binomial distribution B(n,p)can be approximated by a Poisson distribution provided
  • n is large ( typically >  50 )
  • p is small
  • np is approximately less than 5
• The mean of a binomial distribution can be calculated by:
• The Poisson distribution is derived from the binomial distribution for conditions where n is becoming infinitely large and p is becoming infinitely small

Do I need to use continuity corrections?

• No!
• As both the binomial distribution and Poisson distribution are discrete there is no need for continuity corrections

How will I choose which approximation to use?

• When deciding what approximating distribution to use first make sure you know the reason why you cannot find the probability using the original distribution
  • Is the value of n or λ too large?
  • Will it take too long to carry out the calculations?
• Make sure you know what distribution you are approximating from
  • If your distribution is a binomial distribution, you could either use a Poisson or a normal approximation
  • If your distribution is a Poisson distribution, you will use a normal approximation
• Use the conditions for approximations to decide which approximation is appropriate
• Calculate the parameters for the approximating distribution