2.4.2 Approximations of Distributions

Normal Approximation of Poisson

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

A Poisson distribution $X ~ P o (λ)$ can be approximated by a normal distribution $X_{N} ~ N (μ, σ^{2})$ provided
- $λ$ is sufficiently large ( > 15)
Remember that the mean and variance of a Poisson distribution are approximately equal, therefore the parameters of the approximating distribution will be:
- $μ = λ$
- $σ^{2} = λ$
- $σ = \sqrt{λ}$
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

2-4-2-approximations-of-distributions-diagram-1

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
- For example, X = 4 for Poisson can be thought of as $3.5 \leq X_{N} < 4.5$ for normal as every number within this interval rounds to 4
- Remember that for a normal distribution the probability of a single value is zero so $P (3.5 \leq X_{N} < 4.5) = P (3.5 < X_{N} < 4.5)$

Do I need to use continuity corrections?

Yes!
As the Poisson distribution X is discrete and normal distribution X_N is continuous you will need to use continuity corrections
- $P (X = k) \approx P (k - 0.5 < X_{N} < k + 0.5)$
- $P (X \leq k) \approx P (X_{N} < k + 0.5)$
- $P (X < k) \approx P (X_{N} < k - 0.5)$
- $P (X \geq k) \approx P (X_{N} > k - 0.5)$
- $P (X > k) \approx P (X_{N} > k + 0.5)$

How do I approximate a probability?

STEP 1: Find the mean and variance of the approximating distribution
- $μ = σ^{2} = λ$
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
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

Worked example

The number of hits on a revision web page per hour can be modelled by the Poisson distribution with a mean of 40. Find the probability that there are more than 50 hits on the webpage in a given hour.

2-4-2-approximations-of-distributions-we-solution-1

Poisson Approximation of Binomial

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:
- $λ = n p$
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

Worked example

It is known that one person in a thousand who checks a revision website will choose to subscribe. Given that the website received 3000 hits yesterday, find the probability that more than 5 people subscribed.

2-4-2-approximations-of-distributions-we-solution-2

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 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

2-4-2-choosing-approximations-diagram-1

Exam Tip

If you are unsure, use the normal distribution to approximate the Poisson and the Poisson distribution to approximate the binomial as these are the two you will be examined on in Statistics 2. However, you should make sure you know the conditions for the normal approximation to the binomial as this may be tested.

CIE A Level Maths: Probability & Statistics 2

Revision Notes

Normal Approximation of Poisson

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

What are continuity corrections?

Do I need to use continuity corrections?

How do I approximate a probability?

Worked example

Poisson Approximation of Binomial

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

Do I need to use continuity corrections?

Worked example

Choosing the Approximation

How will I choose which approximation to use?

Exam Tip

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Author: Amber