J. K.

SHAH CLASSES Poisson Distribution

10. Poisson Distribution

1. It is a discrete probability distribution where the variable ‘x’ can assume values ‘x’= 0,
1, 2, 3,......∞.

2. This distribution is a limiting case of Binomial Distribution when

(i) n → ∞ i.e. no. of trials become very large
(ii) p → 0, (i.e. probability of success is very small)
(iii) np is finite and constant which is denoted by ‘m’ i.e. np = m or ‫ג‬

3. Poisson Distribution is also known as “Distribution of Improbable Events or

Distribution of Rare Events.

4. Some examples of Poisson Distribution:

(i) No. of telephones calls per minute at a switch board
(ii) The no. of printing mistake per page in a large text.
(iii) The no. of cars passing a certain point in 1 minute
(iv) The emission of radio active (alpha) particles.

5. The conditions under which the Poisson Distribution is used or the condition for
Poisson Model are as follows:
(i) The probability of having success in a very small time interval (t, t + dt) is kt
(where k > 0 and is constant)
(ii) In other words, probability of success in a very small time interval is directly
proportional to time (t).
(iii) The probability of having more than one success in this time interval is very low.
(iv) Statistical independence is assumed i.e. the probability of having success in this
time interval is independent of time ‘t’ as well as of the earlier success.

6. Characteristic or Properties of Poisson Distribution.

(i) Poisson Distribution is uniparametric i.e. it has only one parameter ‘m’ or ‘‫’ג‬
(ii) Mean of distribution = m
(iii) Variance = m
(iv) In poisson distribution mean = variance and hence they are always positive
(v) SD = m
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(vi) Skewness = 1
m

Note 1 Since ‘m’ is always positive Poisson Distribution is always positively

skewed.

Note 2 For large ‘m’ ( 1 ) becomes very small and the Poisson Distribution
m

tends to be symmetrical because skewness approaches zero.

(vii) Kurtosis = 1
m

(viii) The distribution can be either unimodal or bimodal depending on values of m.

Case I : When ‘m’ is not an integer then the distribution is uni-modal and the
value of the mode will be highest integral value contained in ‘m’.

E.g. m = 5.6 then modal value is 5 (greatest integer contained in 5.6)

Case II: When ‘m’ is an integer; the distribution is bimodal and the modal values
are m, m – 1

E.g . if ‘m’ = 4 (an integer, hence the distribution is bimodal and the
modes are 4 and 4 – 1 i.e. 4 and 3)

(ix) Additive Property of Poisson Distribution: If ‘x’ and ‘y’ are two independent
Poisson Variates with parameters(m1) and (m2) respectively then (x + y) will also
follow a Poisson Distribution with parameter (m1 + m2). Symbolically the fact is
expressed as follows: X ~ P (m1)
Y ~ P (m2)
X + Y ~ P(m1 + m2) provided x and y are independent
(x) The probability of ‘x’ no. of success or the p.m.f (Probability Mass Function) of a
Poisson Distribution is given by
x x
f(x)/P(x) = e− m. m
x!
or e− λ. λ
x!
( ‫ = ג‬m)

where x = desired no. of success.

(‫ = ג‬m) Mean = variance = parameter of the distribution

e−m or e−λ is a constant and the value of which can be obtained from the table.

Note: When the parameter ‘m’ is not provided but n and p are provided we shall
use m = np for evaluating the parameter.

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Poisson Distribution – Properties & Problems

1. If a random variable X follows Poisson Distribution, such that P(X = 1) = P(X= 2), then
its mean and variance are:

a) 4, 4

b) 3, 3

c) 2, 2

d) 5, 5

2. A random variable X follows Poisson Distribution, such that P(X = k) = P(X = k + 1),
then its mean and variance is:

a) k – 1, k – 1

b) k + 2, k + 2

c) k + 3, k + 3

d) k + 1, k + 1

3. In a Poisson Distribution P(X = 0) = P(X = 1) = k, the value of “k” is:

a) 1

1
b)
e

c) e2

1
d)
e

4. If X is a Poisson Variate with P(X = 0) = P(X = 1), Then P(X = 2) = ?

a) e

1
b)
2e

2
c)
e

1
d)
e

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5. For a Poisson Variate X, if P(X= 0) = 0.2, then the variance of the distribution is:
a) log e 2

b) log e 4

c) log e 5

d) None of the above

6. The mean of a Poisson Distribution is 0.5, then the ratio of P(X = 3) to P(X = 2) is:
a) 1:6
b) 6:1
c) 1:3
d) 2:5

7. If x is Poisson variety with a parameter 4 find the Mode of the Distribution?

A) 4,2
B) 4,3
C) 4,4
D) None
A random variable X follows Poisson distribution with parameter 4. Find the probability that:
(Given e −4 = 0.0183)

8. P(X = 0)
a) 0.0183
b) 0.15616
c) 0.1952
d) None of the above

9. P(X = 1)
a) 0.1464
b) 0.0732
c) 0.3725
d) None of the above

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10. P(X = 2)

a) 0.3752

b) 0.0732

c) 0.1464

d) 0.6442

11. P(X = 3)

a) 0.1952

b) 0.1529

c) 0.1295

d) 0.2052

12. P(X is at least = 3)

a) 0.7621

b) 0.2671

c) 0.6721

d) None of the above

13. P(X is almost 2)

a) 0.3297

b) 0.2549

c) 0.2379

d) None of the above

14. P(X is more than 3)

a) 0.6669

b) 0.5596

c) 0.5779

d) 0.5669
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15. P(X is 2 or more than 2)

a) 0.2497

b) 0.4297

c) 0.4081

d) 0.9085

16. P (3< x < 5)

a) 0.5219

b) 0.1952

c) 0.2954

d) 0.3459

17. P (3 ≤ x < 5)

a) 0.3904

b) 0.2904

c) 0.1904

d) 0.0904

18. P (3 < x ≤ 5)

a) 0.3595

b) 0.3513

c) 0.4513

d) 0.2549

19. P (3 ≤ x ≤ 5).

a) 0.5987

b) 0.4598

c) 0.4665

d) 0.5465
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J. K. SHAH CLASSES Poisson Distribution

20. The Standard Deviation of the given Poisson Distribution is:

a) 1
b) 2
c) 3
d) 4

21. The standard deviation of a Poisson Variate is 3. Find the probability that x = 2.
[ e −3 = 0.0498]
a) 0.2241
b) 0.1422
c) 0.2142
d) 0.2214

Between 4 and 5 PM, the average number of phone calls per minute coming into the
switchboard of the company is 3. Find the probability that in one particular minute there will
be: (Given e −3 = 0.0498)
22. No phone call
a) 0.0498
b) 0.0598
c) 0.4598
d) 0.4587

23. Exactly 2 phone calls

a) 0.1422
b) 0.2214
c) 0.2251
d) 0.2241
It is found that the number of accidents occurring in a factory follows Poisson distribution
with a mean of 2 accidents per week. (Given e −2 = 0.1353)
24. Find the probability that no accident occurs in a week
a) 0.531
b) 0.315
c) 0.135
d) None of the above

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25. Find the probability that the number of accident in a week exceeds 2.

a) 0.325

b) 0.523

c) 0.352

d) None of the above

The number of accidents attributed in a year to the taxi drivers in a city follows Poisson
Distribution with mean 3. Out of 1000 taxi drivers, find approximately: ( e −1 = 0.3879, e −2 =
0.1353, e −3 = 0.0498)

26. The number of taxi drivers with no accident in a year.

a) 45

b) 60

c) 50

d) 75

27. The number of taxi drivers with more than 3 accidents in a year

a) 303

b) 353

c) 453

d) 403

28. A radioactive source emits on the average 2.5 particles per second. Calculate that 2 or
more particles will be emitted in an interval of 4 seconds.

a) 11e −10

b) 1 − 10e−10

c) 1 − 11e −10

d) None of the above

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Binomial Approximation to Poisson Distribution

If 3% of the bolts manufactured by the company are defective, what is the probability that in
a sample at 200 bolts: (Given e −6 = 0.00248)
29. 5 bolts will be defective?
a) 0.611
b) 0.015
c) 0.160
d) 0.258

30. None is defective?

a) 0.00248
b) 0.00496
c) 0.00124
d) None of the above

Experience has shown that, as the average, 2% of the airline’s flights suffer a minor
equipment failure in an aircraft. Estimate the probability that the number of minor
equipment failures in the next 50 flights will be(e-1=.3679)

31. 0 (Zero)
a) 0.3679
b) 0.2498
c) 0.3598
d) None of the above

32. At least 2 (Two)

a) 0.2224
b) 0.4424
c) 0.2242
d) None of the above

1/ 5 %
of the blades produced by a blade manufacturing factory turn out to defective. The
blades are supplied in packets of 10. Use Poisson distribution to calculate the number of
packets in a consignment of 100000 packets: ( e −0.02 = 0.9802)

33. Containing no defective

a) 97580
b) 98020
c) 98000
d) 99020
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34. Containing one defective

a) 1900
b) 1978
c) 1987
d) 1960

35. Containing two defectives

a) 15
b) 20
c) 25
d) 35

36. In a company manufacturing toys, it is found that 1 in 500 is defective. Find the
probability that there will be at the most two defectives in a sample of 2000 units.
[Given e-4 = 0.0183]
a) 0.2597
b) 0.3549
c) 0.2549
d) 0.2379

37. If 2% of the items made by a factory are defective. Find the probability that there are 3
defective items in a sample of 100 items.
a) 0.190
b) 0.154
c) 0.180
d) None of the above

38. If the chance of being killed by Flood during a year is 1/3000, use Poisson distribution
to calculate the probability that out of 3000 persons living in a village at least one
would die in flood in a year.

a) e −1

b) e

c) 1 − e −1

d) None of the above

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Miscellaneous Problems
39. A local electric appliance has found from experience the demand for Tube light is
distributed as Poisson distribution with a mean of 4-tube light per week. [Given e-4 =
0.0183] if the shop keeps 6 tubes during a particular week, what is the probability that
the demand will exile the supply during that week
a) 0.1114
b) 0.2224
c) 0.1525
d) 0.1254

40. A car hire firm has 2 cars which is hired out every day. The number of demand per
day for a car follows Poisson distribution with mean 1.20. What is the proportion of
days on which some demand is refused?
(Given e1.20 = 3.32)
a) 0.25
b) 0.3012
c) 0.12
d) 0.03

41. If x is a Poisson Variable such that P(x = 2) = 9 P(x = 4) + 90 P(x = 6), find the mean
and variance of x
a) 2
b) 3
c) 1
d) 4

42. P( x ≤ 2 / x ≥ 1) given E(x) = 2.2 & e(–2.2) = .1108

a) 0.58
b) 0.68
c) 0.70
d) None of the above

43. The administrator of A LARGE AIRPORT IS INTRESTED IN THE NUMBER of aircraft

departure delays that are attributable to inadequate control facilities. A random
sample of 10 aircraft take-off is to be thoroughly investigated. If the true proportion of
such delays in all departures is 0.40, then the probability that 4 of the sample
departures are delayed because of control inadequacies is:
a) 0.2508
b) 0.212
c) 0.152
d) 0.3

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Fitting a Data to Poisson Distribution

44. Fit a Poisson distribution to the following:

Number of Death: 0 1 2 3 4

Frequency: 122 46 23 8 1

Theoretical Aspects

45. Poisson distribution is a _____________ probability distribution.

a) Continuous

b) Discrete

c) Both of a) and b) above

d) Neither a) nor b) above

46. Which one is uni-parametric distribution?

a) Normal Distribution

b) Poisson Distribution

c) Geometric Distribution

d) Binomial Distribution

47. Which one is not a condition of Poisson model?

a) The probability of having success in a small time interval is constant

b) The probability of having success in a small interval is independent of time and

also of earlier success

c) The probability of having success more than one in a small time interval is very
small.

d) None of the above

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48. __________ Distribution is a limiting case of Binomial distribution.

a) Normal Distribution

b) Poisson Distribution

c) Chi-Square Distribution

d) (a) & (b) both

49. Poisson distribution may be

a) Bimodal
b) Uni modal
c) Multi Modal
d) Either a) or b) above and not c)

50. ___________ Distribution is sometimes known as the “distribution of rare events”.

a) Binomial
b) Normal
c) Geometric
d) Poisson

51. In ____________ distribution, mean = variance

a) Chi- Square
b) Normal
c) Poisson
d) Hypergeometric

52. When the number of trials is large and probability in small then the distribution used is:
a) Poisson Distribution
b) F – Distribution
c) t – Distribution
d) Normal Distribution

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53. For a Poisson distribution

a) Standard Deviation and Variance are equal.
b) Mean and Variance are equal.
c) Mean and Standard Deviation are equal.
d) Both a) and b) above

54. Number of radio-active atoms decaying in a given interval of time is an example of

a) Normal distribution

b) Binomial distribution

c) Poisson distribution

d) None of the above

55. In Poisson Distribution, probability of success is very close to

a) 1

b) 0.8

c) 0

d) None of the above

56. Poisson distribution is

a) Always negatively skewed

b) Always positively skewed

c) Always symmetric

d) Symmetric only when m = 2

57. The Poisson distribution tends to be symmetrical if the mean value is

a) Zero

b) Very Low

c) High

d) None of the above

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THEORY ANSWERS:

45 b 51 c 57 c

46 b 52 a

47 a 53 b

48 b 54 c

49 d 55 c

50 d 56 b

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