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(f) Use continuity corrections for discrete random variable LEARNING OUTCOMES At the end of the lesson, students will be able to (g) Use the normal distribution to approximate binomial distribution 2

The Approximation from Binomial Distribution to Normal Distribution to Normal Distribution Under certain circumstances, normal distribution can be used as an approximation to Binomial Distribution with Where This approximation is used when (i) The value of n is large (ii) The value of p is close to 0.5 3

CONTINUITY CORRECTION Suppose a coin tossed 12 times. If we required the probability that there are not more than three head, i.e then we consider So P( X  3) transforms to P( X < 3.5 ), i.e. P( X  3)  P( X < 3.5 )

CONTINUITY CORRECTION 5

CONTINUITY CORRECTION 6

Example The area of rectangle should be considered

The area of rectangle should not be considered

Only the area of rectangle should be considered

Example 1 Write down each probability below after continuity correction 12

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Example 2 Let, use the binomial approximation to find 14

Solution Given, where n = 200, p = 0.3, q=0.7. Normal approximation 15

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Example 3 Find the probability of obtaining between 4 and 6 heads ( inclusive) when tossing a fair coin 12 times, by using a) The binomial distribution b) The normal approximation to the binomial distribution 21

Solution Let X = the number of heads obtained Then, 22

b) Using the normal approximation 23

Example 4 A fair coin is tossed 400 times. Find the probability of getting a) Less than 230 tails b) Exactly 205 tails c) Between 180 and 190 tails solution Let X = the number of tails obtained Then, The normal approximation 24

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The Approximation from Poisson Distribution to Normal Distribution Under certain circumstances, normal distribution can be used as an approximation to Poisson Distribution with This approximation is used when the value of 28

Example 6 29

Solution 30

Example 7 The number of complain received by a telecommunication company has a Poisson distribution with mean 10 complain per day. Find the probability that: (a) there were more than 10 complain received in a day (b) at least 50 complain in a week 31

Solution 32

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Example 8 Assume that the number of s received by a student daily has a Poison Distribution with a mean of 5. a) Determine the probability that the student receives between 5 and 13 s daily. b) If 15 days are randomly chosen, find the probability that the student receives between 5 and 13 s daily for a period of 9 days. 34

c) If 150 days are randomly chosen, use the normal approximation to find the probability that the student receives between 5 and 13 s daily for less than 70 days. 35

Solution Let X= number of s received by a student daily. 36

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38 Let Y= number of students receive between 5 and 13 s daily

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Example 9 The number of asbestos particles in a squared centimeter of dust is found to follow a Poisson distribution with a mean of If a random squared centimeter of dust is analysed, what is the probability that less than 950 particles are found? 40

Solution Using the normal approximation : 41

Example 10 Ten present of tiles produced in a factory are broken before they are packed. If a random sample of 500 tiles is taken, find the probability of getting a) Less than 40 broken tiles b) At least 40 broken tiles c) Between 50 and 56 broken tiles ( inclusive) d) At most 30 broken tiles 42

solution Let X = the number of broken tiles obtained.Then, The normal approximation 43

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Using the Poisson Distribution as an Approximation to the Binomial distribution It is appropriate to use the Poisson distribution as an approximation to the binomial when (i) n is large ( n > 50 ) or / and (ii) p is small ( p < 0. 1 ) In fact, when p = 0.1, n 30, both Poisson and binomial distribution are almost identical. This particular approximation is more accurate when p 0 and n ( for Poisson ) = np ( for binomial) 48

The Approximation from Binomial Distribution to Normal Distribution to Normal Distribution Normal distribution can be used as an approximation to Binomial Distribution with Where q = 1-p This approximation is used when (i) The value of n is large (ii) The value of p is close to

The Approximation from Poisson Distribution to Normal Distribution to Normal Distribution Under certain circumstances, normal distribution can be used as an approximation to Poisson Distribution with This approximation is used when the value of 50

Exercise The discrete random variable X is found to follow a Poisson distribution with a mean of 200. Use the Normal approximation of this Poisson distribution to find : Answer : (a) 0.65 (b)