Approximating One Distribution With Another

What with What?

Approximations are used when one or more of the parameters of a Discrete distribution cannot be found in the tables. Both Binomial and Poisson distributions can be approximated by the Normal. Also the Binomial can be approximated by the Poisson

The Poisson approximated by the Normal

When the value of l is too large to read from the tables, the Normal Distribution is used to approximate it. The two things not to forget are:

• Use Continuity Correction
• The mean and variance of a Poisson are l .
  

The continuity correction compensates for the conversion from a Discrete to a Continuous distribution, so for example, p (X<23) becomes p (X<22.5) because 22 in the Discrete model is approximated by 21.5 to 22.5 in the Continuous one.

 

The Binomial approximated by the Normal

When the value of n in a Binomial Distribution is too large to read from the tables, and p is near 0.5, use the Normal to approximate it. Again remember two things: 

• Use Continuity Correction
• The mean of a binomial is np, the variance is np(1-p). 

Again, we are approximating a Discrete to a Continuous, so convert the numbers! The approximation gets better as n gets larger, and p gets closer to 0.5. If p is closer to 0 or 1, use the Poisson instead.

 

 

The Binomial approximated by the Poisson

When n is too large, and p is near 0 or 1, use the Poisson to approximate it. The value for l is simply found by multiplying n and p. (If the value of p is close to one, then we consider the number of failures instead).

The approximation gets better as n gets larger, and p gets closer to 0 or 1.