S2 Chapter 5: Normal Approximations

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 29th September 2014

RECAP: Poisson Binomial

We saw earlier that conceptually speaking a Poisson Distribution is a Binomial Distribution in disguise. Why?The Poisson Distribution counts the occurrence of a event in a time period. We can divide the time period into a large number of time intervals, each a trial in which the event can occur or not occur (i.e. two outcomes). The count of events is then Binomially Distributed, but as the time intervals become infinitely small (i.e, large and smaller), we got the Poisson Distribution in the limit.

If , then what distribution could we use to approximate this?

Under what conditions is the approximation sufficiently accurate?As becomes large and is small. Rule of thumb is when .

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Calculating probabilities for a Poisson is fine if is small. But if is large, is difficult to calculate. Is there an easier approximation to use?

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Binomial ?

(#successes)

𝑝(𝑥

)

The graph shows the probability function for different Binomial Distributions. Which one resembles another distribution and what distribution does it resemble?When , and is fairly large, it resembles a normal distribution in part because it’s symmetrical. This is not a coincidence – if you do S3 you’ll see this arises due to something called the Central Limit Theorem.

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This chapter is about how we can approximate either the Binomial or Poisson Distribution using the Normal Distribution.

Continuity Corrections

We wish to approximate the Binomial and Poisson distributions using a Normal distribution. One problem is that Binomial/Poisson are discrete whereas the Normal distribution is continuous.We apply something called a continuity correction to approximate a discrete distribution using a continuous one.

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Discrete:

Continuous:

The random variable represents the time to finish a race in a whole number of hours. We’re interested in knowing the probability Alice took 6 hours.How would you represent this time on a number line given hours is discrete? And what about if hours was now considered to be continuous (as )?

We can’t just find when is continuous, because the probability is effectively 0. But would seem a sensible interval to use because any time between 5.5 and 6.5 would have rounded to 6 hours were it discrete.

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Continuity Corrections

If is a discrete variable, and is its continuous equivalent, how would you represent for ?

! A continuity correction is approximating a discrete range using a continuous one.1. If > or <, convert to first.2. Enlarge the range by 0.5.

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Discrete:

Continuous:

𝑋 ≥5𝑌 ≥4.5

Notice the range has been enlarged by an extra 0.5.

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How would represent for ?

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Discrete:

Continuous:

𝑋<9→𝑋 ≤8𝑌 ≤8.5?

Examples

! A continuity correction is approximating a discrete range using a continuous one.1. If > or <, convert to first.2. Enlarge the range (at each end) by 0.5.

Discrete Continuous

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Binomial NormalTo approximate a Binomial Distribution as a Normal Distribution, we just copy over the mean and variance of the Binomial to the Normal. Sorted!

𝑋 𝐵 (𝑛 ,𝑝 ) approximate

if is large, close to 0.5

𝑌 𝑁 (𝑛𝑝 ,𝑛𝑝 (1−𝑝 ) )? ?

!𝑥

𝑝 (𝑥 )

𝑥

𝑓 (𝑥 )

Thus

Bro Tip: It’s a common error to accidentally forget to square root the variance to get the standard deviation – we need it in calculation of probabilities.

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Example

.a) Use tables to find exactly. 0.25b) Use a normal approximation to estimate .

Test Your Understanding:. Use a normal approximation to estimate .

. Use a normal approximation to estimate .

Note that < vs doesn’t matter at this point as is continuous thus the probability is not affected.

We’ve standardised as per S1 to get .

Recall that either changing the direction of the inequality or changing the sign gives us We need < and a positive value to use table.

We both changed > to < and changed sign, so two cancel each other out.

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Exercise 5B

. Use a suitable approximation to estimate:

. Use a suitable approximation to estimate:

In a multiple choice test there are 4 possible answers to each question. Given that there are 60 questions on the paper, use a suitable approximation to estimate the probability of getting more than 20 questions correct if the answer to each question is chosen at random from the 4 available choices for each question.

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Poisson NormalAgain, we want the same mean and variance for the Normal as the original Poisson.

𝑋 𝑃𝑜 (𝜆 ) approximate

if is large

𝑌 𝑁 (𝜆 ,𝜆 )Thus ?

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Test Your Understanding

!

. Determine

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Exam Example

Edexcel S2 Jan 2012

𝑿 𝑷𝒐 (𝟓 )

(300 hits per hour is 50 hits per 10 mins)?

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All Approximations Summarised!This handy diagram (found in your textbook) summarises all approximations very nicely.This will help you decide when the exam question says “Using a suitable approximation…”

Discrete Continuous

Binomial

Normal

Poisson

( large, close to 0.5)

large

(Large , small ) 𝜆=𝑛𝑝

𝜇=𝜆

𝜇=𝑛𝑝

Continuity correction needed if line crossed/

Is ?

YESNO

𝐵→𝑁𝐵→𝑃𝑜𝑃𝑜→𝑁

On the next few slides, read the question and vote with your

diaries what the original distribution is and what the new

approximated one is.

𝐵→𝑁𝐵→𝑃𝑜𝑃𝑜→𝑁

Q: A spinner is designed to land on red 10% of the time. Use a suitable

approximation to estimate the probability of fewer than 4 red in 60 turns of the

spinner.Because , which is less than 10.

𝐵→𝑁𝐵→𝑃𝑜𝑃𝑜→𝑁

Q: A spinner is designed to land on red 10% of the time. Use a suitable

approximation to estimate the probability of more than 20 red in 150 turns of the

spinner.Because . We try see if a Poisson approximation is appropriate first. Even if is not close to 0.5, we use the normal approximation anyway!

𝐵→𝑁𝐵→𝑃𝑜𝑃𝑜→𝑁

Q: A Royal Mail processing centre receives on average 1200 letters a minute. Use a suitable approximation to work out the probability that it receives more than

1500 in a given minute.

Exercises

In a village, power cuts occur randomly at a rate of 3 per year.a) Find the probability that in any given year

there will be (i) exactly 7 power cuts, (ii) at least 4 power cuts. (5)(i) (ii)

b) Use a suitable approximation to find the probability that in the next 10 years the number of power cuts will be less than 20. (6)

The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.(c) Use a suitable approximation to estimate the probability that the estate agent receives a bonus.

Edexcel S2 Jan 2013 Q2

Edexcel S2 May 2012 Q4

Defects occur at random in planks of wood with a constant rate of 0.5 per 10 cm length. Jim buys a plank of length 100 cm. a. Find the probability that Jim’s plank

contains at most 3 defects. (2)

b. Shivani buys 6 planks each of length 100 cm. Find the probability that fewer than 2 of Shivani’s planks contain at most 3 defects. (5)

c. Using a suitable approximation, estimate the probability that the total number of defects on Shivani’s 6 planks is less than 18. (6)

Edexcel S2 May 2011 Q5

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