J. K. SHAH CLASSES Binomial Distribution
9. Binomial Distribution
THEORETICAL DISTRIBUTION (Exists only in theory) 1. Theoretical Distribution is a distribution where the variables are distributed according to some definite mathematical laws. 2. In other words, Theoretical Distributions are mathematical models; where the frequencies / probabilities are calculated by mathematical computation. 3. Theoretical Distribution are also called as Expected Variance Distribution or Frequency Distribution 4. THEORETICAL DISTRIBUTION
DISCRETE CONTINUOUS
BINOMIAL POISSON Distribution Distribution
NORMAL OR Student’s ‘t’ Chi square Snedecor’s GAUSSIAN Distribution Distribution F Distribution Distribution A. Binomial Distribution (Bernoulli Distribution) 1. This distribution is a discrete probability Distribution where the variable ‘x’ can assume only discrete values i.e. x = 0, 1, 2, 3,...... n 2. This distribution is derived from a special type of random experiment known as Bernoulli Experiment or Bernoulli Trials , which has the following characteristics (i) Each trial must be associated with two mutually exclusive & exhaustive outcomes – SUCCESS and FAILURE . Usually the probability of success is denoted by ‘p’ and that of the failure by ‘q’ where q = 1 p and therefore p + q = 1. (ii) The trials must be independent under identical conditions. (iii) The number of trial must be finite (countably finite). (iv) Probability of success and failure remains unchanged throughout the process.
: 442 : J. K. SHAH CLASSES Binomial Distribution
Note 1 : A ‘trial’ is an attempt to produce outcomes which is neither sure nor impossible in nature.
Note 2 : The conditions mentioned may also be treated as the conditions for Binomial Distributions.
3. Characteristics or Properties of Binomial Distribution (i) It is a bi parametric distribution i.e. it has two parameters n & p where n = no. of trials p = probability of success. (ii) Mean of distribution is np. (iii) Variance = npq (iv) Mean is greater than variance always i.e. np > npq.
(v) SD = npq (vi) Maximum variance is equal to (n/4) −−− (vii) Skewness = q p npq Binomial Distribution may be Symmetrical or Asymmetrical (i.e. skewed) where q > p; its positively skewed and when q < p its negatively skewed. When q = p = 0.5 skewness is equal to zero. In such a case, the distribution is said to be symmetrical. −−− (viii) Kurtosis = 1 6pq npq (ix) Binomial Distribution may be Uni Modal or Bi Modal depending on the values of the parameters n & p.
Case I : When (n + 1).p is not an integer the distribution is uni modal and the greatest integer contained in (n+1) p is the value of the mode. E.g. n = 6; p = 1/3; find modal value.
Solution : (n + 1)p = (6 + 1) x 0.3
= 7 x 0.3 = 2.1 which is not an integer. Hence the given distribution is unimodal and the value of mode is equal to 2 (Greatest integer integral value in 2.1)
: 443 : J. K. SHAH CLASSES Binomial Distribution
Case II: When (n + 1)p is an integer; the distribution is bi modal and the modal values are (n+1)p and (n+1)p – 1 respectively. E.g. n = 7 and p = 0.5; find mode or modes.
Solution : (n +1)p = (7 + 1)p
= 8(0.5)
= 4. Which is an integer.
Hence the two modes are :4 & (4 1) =3
(x) Additive Property of Binomial Distribution: If ‘x’ and ‘y’ are two independent
binomial variates with parameters(n 1, p) and (n 2, p) respectively, then x + y
will also follow a binomial distribution with parameters {(n 1 + n 2), p} Symbolically the fact is expressed as follows:
X ~ B (n 1, p)
Y ~ B (n 2, p)
X + Y ~ B(n 1 + n 2, p) (xi) The method applied for fitting a binomial distribution to a given set of data is called “Method of Moments”. 4. The distribution is called Binomial as the probabilities are given by the series (q + p) n 5. The probabilities of ‘x’ number of success or the p.m.f (Probability Mass Function) of a Binomial Distribution is given by :
n pxqn−−−x f(x) / p(x) = cx where, p = probability of success q = probability of failure=(1 p) ‘x’= no. of success And (n –x) = no. of failures
Note 1: Sum of powers of p and q will always add up to ‘n’ irrespective of no. of success.
Note 2: There are(n + 1) possible value of ‘x’ i.e. x = { 0,1,2,3,..... ,n}
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CONCEPT OF UNIFORM DISTRIBUTION (DISCRETE VARIANCE)
1. If a discrete random variable ‘x’ assumes n possible values namely x 1 x2, .... x n with equal probabilities, then the probability of its taking any particular value is always constant and is equal to (1/n). The p.m.f (Probability Mass Function) of such
distribution is given by f(x) = 1/n where x = x 1 x2, .... x n. These distributions are known as Uniform Distribution because the probability is uniform for all values of x. e.g. Probability Distribution of the no. of points in a throw of a die. x 1 2 3 4 5 6 p 1 1 1 1 1 1 6 6 6 6 6 6
2. Mean of Uniform Distribution is : n +++ 1 2
Discrete Probability Distribution
1. A random variable X has the following distribution:
X (= x i) 0 1 2 3
2 3 0 1 P(X = x ) i 6 6 6 6
The mean and variance of X are:
a) 1, 2
b) 1, 3
c) 1, 1
d) 3, 1
2. A random variable X takes the values 0, 1 and 2. If P(X = 1) = P(X = 2) and P(X = 0) = 0.4, then the mean of the random variable X is:
a) 1.1
b) 0.9
c) 0.8
d) 0.4
: 445 : J. K. SHAH CLASSES Binomial Distribution
Uniform Distribution 3. A discrete random variable x follows uniform distribution and takes only the values 6, 9, 10, 11, 13. Find P(x = 12). a) 0 b) 4/5 c) 1/5 d) 3/5
4. A discrete random variable x follows uniform distribution and takes only the values 6, 8, 12, 11, 17. Find P(x ≤ 12). a) 1/5 b) 3/5 c) 4/5 d) None of the above
5. A discrete random variable x follows uniform distribution and takes only the values 6, 8, 10, 12, 18. Find P(x < 12). a) 1/5 b) 3/5 c) 4/5 d) None of the above
6. A discrete random variable x follows uniform distribution and takes only the values 5, 7, 12, 15, 18. Find P(x > 10). a) 2/5 b) 3/5 c) 4/5 d) None of the above
7. If a discrete random variable x follows uniform distribution and assumes only the values 8, 9, 11, 15, 18, 20. Then find P(|x – 14| < 5). a) 1 b) ½ c) 2/3 d) 1/3
: 446 : J. K. SHAH CLASSES Binomial Distribution
Properties of Binomial Distribution
8. A random variable x follows Binomial distribution with mean 2 and variance 1.2. Then the value of n is:
a) 3
b) 5
c) 7
d) 6
9. If in a Binomial distribution np = 9 and npq = 2.25, then q is equal to:
a) 0.75
b) 0.25
c) 0.10
d) 0.45
10. If in a Binomial distribution mean 20; S.D. = 4, then p is equal to:
a) 1/5
b) 2/5
c) 3/5
d) 4/5
11. A student obtained the following results: For the binomial distribution mean = 4, variance = 3. Comment on the accuracy of his results.
a) Correct results, with n = 12
b) Correct results, with n = 16
c) Wrong calculations, as q > 1
d) None of the above
: 447 : J. K. SHAH CLASSES Binomial Distribution
12. A student obtained the following results: For the binomial distribution mean = 7, variance = 11. Comment on the accuracy of his results.
a) Correct results, with n = 16
b) Correct results as variance is greater than mean
c) Wrong calculations as variance can’t be greater than mean
d) None of the above
13. Moi obtain the following results: The mean of a Binomial distribution is 3 and the standard deviation is 2. Comment on the accuracy of his calculations.
a) Wrong calculations as variance cannot be more than the mean.
b) Correctly done
c) Correct calculations with n = 100
d) Nothing can be said
14. An unbiased dice is tossed 500 times. The standard deviation of the number of ‘sixes’ in these 500 tosses are:
a) 50/6
b) 51/6
c) 15/6
d) None of these
15. Mean =10, SD= 5 , Mode =?
a) 10
b) 12
c) 9
d) 8 : 448 : J. K. SHAH CLASSES Binomial Distribution
Symmetrical Binomial Distribution 6 coins are tossed. Find the probability of getting
16. All heads a) 1/64 b) 15/64 c) 22/64 d) 5/16
17. No heads a) 15/64 b) 22/64 c) 5/16 d) 1/64
18. Two heads a) 15/64 b) 22/64 c) 5/16 d) 31/64
19. Three heads a) 15/64 b) 22/64 c) 5/16 d) 1/64
20. Two or more heads a) 15/64 b) 22/64 c) 5/16 d) 57/64
21. At most two heads a) 15/64 b) 22/64 c) 5/16 d) 15/22
: 449 : J. K. SHAH CLASSES Binomial Distribution
Binomial Distribution – Applied Problems
22. A machine produces 2% defectives on an average. If 4 articles are chosen randomly. What is probability that there will be exctly 2 defective articles? a) 0.0023 b) 0.115 c) 0.325 d) 0.452 The probability that a college student will be graduate is 0.4.Determine the probability that out of 5 students 23. One. is a graduate a) 0.08 b) 0.26 c) 0.92 d) 0.29
24. At least one will be a graduate a) 0.26 b) 0.08 c) 0.29 d) 0.92
25. The binomial distribution with mean = 20 and sd = 4 is: a) (1/4 + 4/5) 100 b) (4/5 + 1/5) 100 c) (4/5 + 1/5) 50 d) None of the above
Assume the probability that bomb dropped from an aeroplane will strike a target is 1/5. If 6 bombs are dropped, find the probability that 26. Exactly 2 will strike the target a) 698/3125 b) 768/3125 c) 2304/3125 d) 1/2
: 450 : J. K. SHAH CLASSES Binomial Distribution
27. At least 2 will strike the target. a) 4536/15625 b) 2596/3125 c) 6354/15625 d) 5385/15625
28. The overall percentage of failures in a certain examination is 60. What is the probability that out of a group of 6 candidates at least 5 passed the examination? a) (0.4) 6
b) (0.4)3 (0.6) 3
c) (0.6)5+ (0.6) 6
d) 4(0.4) 5
29. A random variable follows binomial distribution with mean4 and S.D. = 2. Find the probability of assuming non zero value of the variant. a) (0.5)
b) (0.5) 8
c) 1− (0.5) 8
d) (0.4)
30. If 15 dates are selected at random, then the probability of getting two Sundays is: a) 0.29 b) 0.39 c) 0.49 d) 0.23
31. Assuming that half the population is vegetarian and each of 128 investigators takes a sample of 10 individuals to see whether they are vegetarian. How many investigators would you expect to report that 2 people or less vegetarians. a) 5 b) 7 c) 9 d) 11 : 451 : J. K. SHAH CLASSES Binomial Distribution
An experiment succeed twice as often as it fails. What is the probability that in the next 5 trials, there will be: 32. 3 successes. a) 77 / 243 b) 81 / 243 c) 80 / 243 d) None of the above
33. At least 3 successes. a) 80 / 243 b) 192 / 243 c) 77 / 243 d) None of the above
34. An experiment succeeds twice as many times as it fails. Find the chance that in 6 trials, there will be at least 5 successes. a) 256 / 729 b) 625 / 729 c) 519 / 729 d) None of the above
An unbiased cubic dice tossed 4 times. What is the probability of obtaining:
35. No. six. a) 630 / 1296 b) 650 / 1296 c) 526 / 1296 d) 625/1296
36. At least one six. a) 671 / 1296 b) 76 / 1296 c) 625 / 1296 d) 645 / 1296
37. All odd numbers.
a) 1 / 4 b) 1 / 12 c) 15 / 16 d) 1/ 16
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If a dice thrown six times, calculate the probability that: 38. A score of 3 or less occurs on exactly two throws. a) 15 / 64 b) 32 / 66 c) 16 / 64 d) None of the above
39. A score of more than two occurs on exactly three throws. a) 16 / 729 b) 160 / 729 c) 13 / 729 d) None of the above
40. A score of 5 or less occurs at least once. a) 46655 / 46666 b) 46655 / 46656 c) 1 / 2 d) None of the above
Rainfalls on an average 12 days in a month of 30 days. Find the probability that in a given week: 41. The any 4 days are fine, remaining wet.
a) (0.6)4 (0.4) 3
b) 35.(0.6)4 (0.4) 3
c) 35.(0.6)3 (0.4) 4
d) (0.6)3 (0.4) 4
42. 3 days are raining.
a) (0.6)4 (0.4) 3
b) 35.(0.6)4 (0.4) 3
c) 35.(0.6)3 (0.4) 4
d) (0.6)3 (0.4) 4
: 453 : J. K. SHAH CLASSES Binomial Distribution
A man takes a step forward with a probability 0.6 and a step backward with a probability of 0.4. Find the probability that at the end of 11 steps, the man is:
43. One step ahead of starting point.
a) (0.6)6 (0.4) 5
b) 462.(0.6)6 (0.4) 5
c) 462.(0.6)5 (0.4) 6
6 5 d) 462. 25
44. One step behind of starting point.
a) (0.6)6 (0.4) 5
b) 462.(0.6)6 (0.4) 5
c) 462.(0.6)5 (0.4) 6
6 5 d) 462. 25
45. One step away of starting point.
a) (0.6)6 (0.4) 5
b) 462.(0.6)6 (0.4) 5
c) 462.(0.6)5 (0.4) 6
6 5 d) 462. 25
46. If x and y are 2 independent binomial variable with parameters 6 and ½, 4 and ½ respectively, what is P(x + y ≥ 1)?
a) 1023/1024
b) 1056/1923
c) 1234/2678
d) None of the above
: 454 : J. K. SHAH CLASSES Binomial Distribution
Calculation of Parameters 47. A binomial random variable x satisfies the relation 9P(x = 4) = P(x =2) when n = 6. Find the value of the parameter ‘P’? a) 1 / 2 b) 1 / 3 c) 1 / 4 d) 1 / 5
48. In a binomial distribution consisting of 5 independent trials, probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. Find the parameter ‘P’ of the distribution? a) 1 / 3 b) 1 / 4 c) 2 / 5 d) 1 / 5
49. In a binomial distribution with 6 independent trials, the probability of 3 and 4 successes are found to be 0.2457 and 0.0819 respectively. Find the parameter ‘P’ of the binomial. a) 3 / 13 b) 4 / 13 c) 5 / 13 d) 6 / 13 Fitting of Bionomial Distributrion(Method of Moment) 50. Fit a Binomial distribution to the following data: x: 0 1 2 3 4 5 f: 3 6 10 8 3 2 Theory – Uniform Distribution
51. The probability distribution whose frequency function f(x) = 1/n, x = x 1, x 2,…,x n is known as: a) Binomial distribution b) Poisson distribution c) Normal distribution d) Uniform distribution
52. Probability Function is known as a) Frequency Function b) Continuous Function c) Discrete Function d) None : 455 : J. K. SHAH CLASSES Binomial Distribution
53. The no. of points obtained in a single throw of an unbiased die follows a) Binomial Distribution b) Poisson Distribution c) Uniform Distribution d) None
54 The no. of points obtained in a single throw of an unbiased, die has frequency function a) f(x) = 1/4 b) f(x) = 1/5 c) f(x) = 1/6 d) None
55. In uniform distribution random variable x assumes n values with a) equal probability b) unequal probability c) zero d) None
56. If for any distribution f(x) = 1/n, then mean of the distribution would be:
a) n/2
b) (n + 1)/2
c) (n – 1)/2
d) None of the above
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Theory – Properties of Binomial Distribution
57. A theoretical probability distribution. a) Does not exit b) exists only in theory c) exists in real life d) None of the above 58. Binomial distribution is a:
a) Discrete Probability Distribution
b) Continuous Probability Distribution
c) Both a) and b) above
d) Neither a) nor b) above
59. An important discrete probability distribution is:
a) Normal Distribution
b) Binomial Distribution
d) None of the above
60. The important characteristic(s) of Bernoulli trials is:
a) Trials are independent
b) Each trial is associated with just two possible outcomes.
c) Trials are infinite
d) Both a) and b) above
61. An example of a bi parametric discrete probability distribution is
a) binomial distribution
b) poisson distribution
c) normal distribution
d) both (a) and (b)
: 457 : J. K. SHAH CLASSES Binomial Distribution
62. In Binomial distribution ‘n’ means
a) Number of trials of the experiment
b) Number of success.
c) The probability of getting success.
d) None of the above
63. In Binomial distribution ‘p’ denotes probability of: a) Success b) Failure c) Both a) and b) above d) None of the above
64. The mean of a Binomial distribution with parameter n and p is: a) n (1 – p) b) np (1 – p) c) np d) n
65. Variance of Binomial distribution is a) np b) npq c) nq d) pq
66. The mean of binomial distribution is : a) Always more than its variance b) Always equal to its standard deviation c) Always less than its variance d) Always equal to its variance.
: 458 : J. K. SHAH CLASSES Binomial Distribution
67. The maximum value of the variance of a Binomial distribution with parameters n and p is :
n a) p
n b) 3
n c) 4
n d) 2
68. For a binomial distribution, there may be a) one mode b) two mode c) zero mode d) (a) or (b)
69. Binomial distribution is symmetrical if: a) p > q b) p = q c) p < q d) p ≠ q
70. When ‘p’ = 0.5, the binomial distribution is: a) Asymmetrical b) Symmetrical c) Both a) and b) above d) Neither a) nor b) above
71. A Binomial distribution is: a) Never negatively skewed b) Never positively skewed c) Never symmetrical d) Symmetrical when p = 0.5
: 459 : J. K. SHAH CLASSES Binomial Distribution
72. For a binomial distribution, mean and mode a) Are always equal b) Are never equal c) are equal when q = 0.50 d) Do not always exist.
73. When ‘p’ is larger than 0.5, the binomial distribution is a) Asymmetrical b) Symmetrical c) Both d) None
74. The probability mass function of binomial distribution is given by : a) f(x) = p x qn+x n x n x b) f(x) = Cx p q n x n x c) f(x) = Cx q p n n+x x d) f(x) = Cx p q
75. For n independent trials in Binomial distribution, the sum of the powers of p and q is always n, whatever be the number of successes. a) True b) False c) both of a) and b) above d) None of the above
76. When there are a fixed number of repeated trials of any experiment under identical conditions for which only one of two mutually exclusive outcome, success or failure can result in each trial then a) Normal Distribution b) Binomial Distribution c) Poisson Distribution d) None is used
77. The results of ODI matches between India and Pakistan follows: a) Binomial distribution b) Poisson distribution c) Normal distribution d) (b) or (c)
: 460 : J. K. SHAH CLASSES Binomial Distribution
78. In a Binomial distribution if n is infinitely large, the probability p of occurrence of event is close to 0 and q is close to 1 then binomial distribution follows to Poisson distribution. a) 0, 1 b) 1,1 c) 1,0 d) None of the above
THEORY ANSWERS:
51 d 61 a 71 d 52 a 62 a 72 c 53 c 63 a 73 a 54 c 64 c 74 b 55 a 65 b 75 a 56 b 66 a 76 b 57 b 67 c 77 a 58 a 68 d 78 a 59 b 69 b 60 d 70 b
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