Presentation on theme: "Poisson approximation to a Binomial distribution"— Presentation transcript:
1 Poisson approximation to a Binomial distribution The Poisson distribution is often used as an approximation to the Binomial distribution for large n and small p, as the Poisson probabilities are easier to calculate. The Poisson approximation to a Binomial distribution may be made provided n is large. In addition p must be quite small, otherwise the two distributions would not exhibit similar amounts of positive skew.
2 Poisson approximation to a Binomial distribution n should be quite large n > 50p should be quite small p < 0.1np should not be too big np < 5X ~ Bin(n, p) approximated byPo(np) where = np.
3 Example 1A factory packs bolts in boxes of 500. The probability that a bolt is defective is Find the probability that a box contains 2 defective bolts.
4 Solution Let X be the r.v. ‘the number of defective bolts in a box’. This is a Binomial distribution, with n = 500 and p =So X ~ Bin( 500, )
5 Method 1: Using the Binomial distribution P(X = x ) = nCx.px.qn-x, x = 0, 1, 2, …….,n.We have n = 500, p = and q = 0.998, so P(X = 2 ) = 500C2.(0.002)2.(0.998)498(500 499)/(2) (0.002)2(0.998)498 = (3 s.f.)
6 Method 2:Since n is large and p is small, we use the Poisson distribution approximationThe parameter = np = 500 = 1So x ~ Po(1) and P(x = r ) =P(X = 2) = e-11/2! = (3 s.f.)Note: the answer agree to 3 S.F. and the calculations were much easier in method 2.
7 Example 2Find the probability that at least two double sixes are obtained when two dice are thrown 90 times.SolutionThrow two dice, P( double 6 ) = 1/6 1/6 = 1/36.Let X be the r.v. ‘the number of double sixes obtained when two dice are thrown 90 times’, then X ~ Bin( 90, 1/36 )and np = 90 1/36 = 2.5
8 Using the Poisson approximation, X ~ Po( 2.5 )and P(X 2) = 1 – P(x 1)= 1- [ e-2.5 ( /1 )]= (3 s.f)Tables
9 Example 3In a large town, one person in 80, on the average, has blood of type X. If 200 blood donors are taken at random, find an approximation to the probability that they include at least five persons having blood of type X.How many donors must be taken at random in order that the probability of including at least one donor of type X shall be 0.9 or more?
10 SolutionLet Y be the random variable 'the number of blood donors of type X'.Then Y ~ Bin (n, p) where n = 200 , p = 1/80We are required to find P (Y 5) and since the criteria for using the Poisson approximation to the binomial distribution are satisfied, this can be approximated toY ~ Po () where = np = 2.5
12 For the second part, we still have p = 1/80 but it is the value of n (sample size) that is unknown, and we are looking for n such that P (at least one donor has type X blood) is 0.9 or more;i.e. P (X 1) 0.9using the Poisson approx again = np = n/80and so P (Y 1) is equivalent to
13 1 – P (Y = 0) 0.9 1 – e– 0.9 1 – e–n/80 0.9i.e. e–n/80 e n/80 1/0.1 e n/80 10take logs to base e;log e n/80 loge10n/80 giving n (80)(2.30) giving: n So we need to take at least 185 donors in order that the probability of including at least one donor of type X is 0.9 or more.
14 ExercisesQuestion1In a certain manufacturing process the proportion of defective articles being produced is 2%. In a batch of 300 articles, find the probability that:(a) there are fewer than 2 defectives(b) there are exactly 4 defectives. Answer
15 Question 2A medical practice screens a random sample of 250 of its patients for a certain condition which is present in 1.5% of the population. Find the probability that they obtained(a) no patients with the condition(b) at least two patients with the condition.Answer
16 Question 3An experiment involving two fair dice is carried out 180 times. The dice are placed in a container, shaken and the number of times a double six is obtained is recorded. Find the probability that the number of times a double six is obtained is:(a) once(b) twice(c) at least three.Answer