Approximating the Binomial Distribution (Edexcel International A Level (IAL) Maths: Statistics 2): Revision Note

Normal Approximation of Binomial

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

• A binomial distribution $X~\mathrm{B}(n,p)$ can be approximated by a normal distribution $X_N~\mathrm{N}(\mu ,{\sigma }^2)$  provided

  • n is large

  • p is close to 0.5

• The mean and variance of a binomial distribution can be calculated by:

  • $\mu =np$

  • ${\sigma }^2=np(1-p)$

Why do we use approximations?

• If there are a large number of values for a binomial distribution there could be a lot of calculations involved and it is inefficient to work with the binomial distribution

  • These days calculators can calculate binomial probabilities so approximations are no longer necessary

  • However it is easier to work with a normal distribution

    • You can calculate the probability of a range of values quickly

    • You can use the inverse normal distribution function (most calculators don't have an inverse binomial distribution function)

Do I need to use continuity corrections?

• Yes!

• As the binomial distribution is discrete and normal distribution  is continuous you will need to use continuity corrections

• $P(X=k)\approx P(k-0.5<X_N<k+0.5)$

• $P(X\le k)\approx P(X_N<k+0.5)$

• $P(X<k)\approx P(X_N<k-0.5)$

• $P(X\ge k)\approx P(X_N>k-0.5)$

• $P(X>k)\approx P(X_N>k+0.5)$

How do I approximate a probability?

• STEP 1: Find the mean and variance of the approximating distribution

  • $\mu =np$

  • ${\sigma }^2=np(1-p)$

• STEP 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

    • Find the standard normal probability and use the table of the normal distribution

• The probability will not be exact as it is an approximate but provided n is large and p is close to 0.5 then it will be a close approximation

Poisson Approximation of Binomial

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

• A binomial distribution X~B(n, p)can be approximated by a Poisson distribution $X_p~Po\left(\lambda \right)$ provided

  • n is large ( typically >  50 )

  • p is small

• The mean of a binomial distribution can be calculated by:

  • $\lambda =np$

• 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

Choosing the Approximation

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 (if p is small) or a normal approximation (if p is close to 0.5)

  • 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