Approximating the Binomial to the Poisson distribution

2 The random variable \(Y\) has the distribution \(\mathrm { B } ( 140,0.03 )\). Use a suitable approximation to find \(\mathrm { P } ( Y = 5 )\). Justify your approximation.

1. (a) Write down the conditions under which the Poisson distribution can be used as an approximation to the binomial distribution.

The probability of any one letter being delivered to the wrong house is 0.01 On a randomly selected day Peter delivers 1000 letters.
(b) Using a Poisson approximation, find the probability that Peter delivers at least 4 letters to the wrong house. Give your answer to 4 decimal places.

4. The random variable \(X \sim \mathrm {~B} ( 150,0.02 )\). Use a suitable approximation to estimate \(\mathrm { P } ( X > 7 )\).

1. The discrete random variables \(W , X\) and \(Y\) are distributed as follows

$$W \sim \mathrm {~B} ( 10,0.4 ) \quad X \sim \operatorname { Po } ( 4 ) \quad Y \sim \operatorname { Po } ( 3 )$$

1. Explain whether or not \(\mathrm { Po } ( 4 )\) would be a good approximation to \(\mathrm { B } ( 10,0.4 )\)
2. State the assumption required for \(X + Y\) to be distributed as \(\operatorname { Po } ( 7 )\) Given the assumption in part (b) holds,
3. find \(\mathrm { P } ( X + Y < \operatorname { Var } ( W ) )\)

6 In restaurant \(A\) an average of 2.2\% of tablecloths are stained and, independently, in restaurant \(B\) an average of 5.8\% of tablecloths are stained.

1. Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson approximation to find the probability that a total of more than 2 tablecloths are stained.
2. Random samples of \(n\) tablecloths are taken from each restaurant. The probability that at least one tablecloth is stained is greater than 0.99 . Find the least possible value of \(n\).

2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.

1. State the exact distribution of \(X\), the number of births in the sample which have the mutation.
2. Explain why \(X\) has, approximately, a Poisson distribution.
3. Use a Poisson approximating distribution to find
   (A) \(\mathrm { P } ( X = 1 )\),
   (B) \(\mathrm { P } ( X > 4 )\).
4. Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
5. Use this Normal approximating distribution to
   (A) find the probability that there are at least 90 births which have the mutation,
   ( \(B\) ) find the least value of \(k\) such that the probability that there are at most \(k\) births with this mutation is greater than 5\%.

5 On a particular night, the number of shooting stars seen per minute can be modelled by the distribution \(\operatorname { Po(0.2). }\)

1. Find the probability that, in a given 6 -minute period, fewer than 2 shooting stars are seen.
2. Find the probability that, in 20 periods of 6 minutes each, the number of periods in which fewer than 2 shooting stars are seen is exactly 13 .
3. Use a suitable approximation to find the probability that, in a given 2-hour period, fewer than 30 shooting stars are seen.

5

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 300,0.01 )\). Use a Poisson approximation to find \(\mathrm { P } ( 2 < X < 6 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\), and \(\mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 )\). Find \(\lambda\).
3. The random variable \(Z\) has the distribution \(\mathrm { Po } ( 5.2 )\) and it is given that \(\mathrm { P } ( Z = n ) < \mathrm { P } ( Z = n + 1 )\).
   (a) Write down an inequality in \(n\).
   (b) Hence or otherwise find the largest possible value of \(n\).

2 The random variable \(X\) has the distribution \(\mathrm { B } ( 400,0.01 )\).

1. Find \(\operatorname { Var } ( 4 X + 2 )\).
2. 1. State an appropriate approximating distribution for \(X\), giving the values of any parameters. Justify your choice of approximating distribution.
   2. Use your approximating distribution to find \(\mathrm { P } ( 2 \leqslant X \leqslant 5 )\).

2

1. The random variable \(W\) has a Poisson distribution.
   State the relationship between \(\mathrm { E } ( W )\) and \(\operatorname { Var } ( W )\).
2. The random variable \(X\) has the distribution \(\mathrm { B } ( n , p )\). Jyothi wishes to use a Poisson distribution as an approximate distribution for \(X\). Use the formulae for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\) to explain why it is necessary for \(p\) to be close to 0 for this to be a reasonable approximation.
3. Given that \(Y\) has the distribution \(\mathrm { B } ( 20000,0.00007 )\), use a Poisson distribution to calculate an estimate of \(\mathrm { P } ( Y > 2 )\).

5

1. Narika has a die which is known to be biased so that the probability of throwing a 6 on any throw is \(\frac { 1 } { 100 }\). She uses an approximating distribution to calculate the probability of obtaining no 6s in 450 throws. Find the percentage error in using the approximating distribution for this calculation.
2. Johan claims that a certain six-sided die is biased so that it shows a 6 less often than it would if the die were fair. In order to test this claim, the die is thrown 25 times and it shows a 6 on only 2 throws. Test at the \(10 \%\) significance level whether Johan's claim is justified.

2 The events organiser of a school sends out invitations to \(\mathbf { 1 5 0 }\) people to attend its prize day. From past experience the organiser knows that the number of those who will come to the prize day can be modelled by the distribution \(\mathbf { B } ( \mathbf { 1 5 0 } , \mathbf { 0 . 9 8 } )\).
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1. Explain why this distribution cannot be well approximated by either a normal or a Poisson distribution. [3]
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2. By considering the number of those who do not attend, use a suitable approximation to find the probability that fewer than 146 people attend. [4]

2 A large hotel has 90 bedrooms. Sometimes a guest makes a booking for a room, but then does not arrive. This is called a 'no-show'. On average \(10 \%\) of bookings are no-shows. The hotel manager accepts up to 94 bookings before saying that the hotel is full. If at least 4 of these bookings are no-shows then there will be enough rooms for all of the guests. 94 bookings have been made for each night in August. You should assume that all bookings are independent.

1. State the distribution of the number of no-shows on one night in August.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. Use a Poisson approximating distribution to find the probability that, on one night in August,
   (A) there are exactly 4 no-shows,
   (B) there are enough rooms for all of the guests who do arrive.
4. Find the probability that, on all of the 31 nights in August, there are enough rooms for all of the guests who arrive.
5. (A) In August there are \(31 \times 94 = 2914\) bookings altogether. State the exact distribution of the total number of no-shows during August.
   (B) Use a suitable approximating distribution to find the probability that there are at most 300 no-shows altogether during August.

2 Javier writes an article containing 52460 words. He plans to upload the article to his website, but he knows that this process sometimes introduces errors. He assumes that for each word in the uploaded version of his article, the probability that it contains an error is 0.00008 . The number of words containing an error is denoted by \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\), giving your answers correct to three decimal places.
   Javier wants to use the Poisson distribution as an approximating distribution to calculate the probability that there will be fewer than 5 words containing an error in his uploaded article.
2. Explain how your answers to part (i) are consistent with the use of the Poisson distribution as an approximating distribution.
3. Use the Poisson distribution to calculate \(\mathrm { P } ( X < 5 )\).

5

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
   2. Given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find an expression for \(\operatorname { Var } ( X )\).
2. 1. The random variable \(Y\) has a binomial distribution with \(\mathrm { E } ( Y ) = 3\) and \(\operatorname { Var } ( Y ) = 2.985\). Find values for \(n\) and \(p\).
   2. The random variable \(U\) has \(\mathrm { E } ( U ) = 5\) and \(\operatorname { Var } ( U ) = 6.25\). Show that \(U\) does not have a binomial distribution.
3. The random variable \(V\) has the distribution \(\operatorname { Po } ( 5 )\) and \(W = 2 V + 10\). Show that \(\mathrm { E } ( W ) = \operatorname { Var } ( W )\) but that \(W\) does not have a Poisson distribution.
4. The probability that, in a particular country, a person has blood group AB negative is 0.2 per cent. A sample of 5000 people is selected. Given that the sample may be assumed to be random, use a distributional approximation to estimate the probability that at least 6 people but at most 12 people have blood group AB negative.
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4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by \(X\).

1. State the distribution of \(X\), giving the values of any parameters.
2. State an approximating distribution for \(X\), giving the values of any parameters.
3. Use this approximating distribution to find each of the following.
   1. \(\mathrm { P } ( X = 4 )\).
   2. \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
4. Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.