Introduction

Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. This range will be between the minimum and maximum statistically possible values, but where the possible value is likely to be plotted on the probability distribution depends on a number of factors, including the distributions mean, standard deviation, skewness and kurtosis. There are many different classifications of probability distributions, including the Poisson, normal and binomial distributions.

In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. The normal distribution is sometimes informally called the bell curve. In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.

In this coursework, the probability of a random variable with specific parameters are determined by using binomial distribution, Poisson distribution and normal distribution. The probability distribution are illustrated graphically. The comparisons between the probability distributions are also made. The impact on the shapes of the graphs by the probability and sample size is investigated.

Methodology

Binomial distribution may be approximated,under certain circumstances,by Poisson distribution or normal distribution.By considering a random variable having binomial distribution,let p=0.05 and n=5,manual calculation is used to obtain probability using binomial

distribution,Poisson distribution and normal distribution.Formula nCx × px ×(1−p)n−x is used to

calculate the binomial distribution probabilities, probabilities of Poisson distribution is obtained

by using formula e−λ (λx)

x ! and normal distribution with formula

x−µσ

, where n is the sample

size, x=0,1,2… n, λ=mean , µ=mean and σ=standard deviation. Continuity correction is carried out by finding the probability between (x-0.5) and (x+0.5) for normal distribution when it is used to estimate the probability of a discrete random variable since it is a continuous random variable.These steps are repeated for n=10, 20 and p=0.1, 0.5 but the calculation is done by using Microsoft Office Excel 2007.The results are tabulated.

The probability distributions are illustrated graphically.27 different graphs are plotted by using Microsoft Office Excel 2007.Column graph is used for binomial distribution and Poisson distribution while scatter graph for normal distribution.

Three graphs for normal distribution of different sample size,n=5,10 and 20 but same probability, that is p=0.05 are combined into one graph.The shapes of distribution are compared and the findings are discussed. The steps is repeated for p=0.1 and p=0.5. The changes in the shapes of distribution of the graph are investigated as n increases.

X Binomial distibution Poisson distribution Normal distribution0 0.7738 0.7788 0.63411 0.2036 0.1947 0.29882 0.02143 0.02434 0.005163 0.001128 0.002028 04 0.00002969 0.0001268 05 0.0000003125 0.000006338 0

Probability distribution for p=0.05

n=5,

Graphs:

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Binomial Distribution

X

Prob

abilit

y

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Poisson Distribution

X

Prob

abilit

y

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normal Distribution

X

Prob

abili

ty

n=10,

X Binomial distibution Poisson distribution Normal distribution0 0.598737 0.606531 0.4266034561 0.315125 0.303265 0.4266034562 0.074635 0.075816 0.0715423 0.010475 0.012636 0.0018484 0.000965 0.00158 6.71548E-065 6.09E-05 0.000158 3.24E-096 2.67E-06 1.32E-05 2.01E-137 8.04E-08 9.4E-07 08 1.59E-09 5.88E-08 09 1.86E-11 3.26E-09 010 9.77E-14 1.63E-10 0

Graphs:

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Binomial Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Poisson Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Normal Distribution

X

Prob

abili

ty

n=20,

X Binomial distribution Poisson distribution Normal distribution0 0.358486 0.367879441 0.2420733341 0.377354 0.367879441 0.3920411082 0.188677 0.183939721 0.2420733343 0.059582 0.06131324 0.0567465174 0.013328 0.01532831 0.0049948415 0.002245 0.003065662 0.0001628066 0.000295 0.000510944 1.93919E-067 3.11E-05 7.2992E-05 8.34851E-098 2.66E-06 9.12399E-06 1.29E-119 1.87E-07 1.01378E-06 7.11E-1510 1.08E-08 1.01378E-07 011 5.17E-10 9.21616E-09 012 2.04E-11 7.68013E-10 013 6.61E-13 5.90779E-11 014 1.74E-14 4.21985E-12 015 3.66E-16 2.81323E-13 016 6.02E-18 0 017 7.46E-20 0 018 6.54E-22 0 019 3.62E-24 0 020 9.54E-27 0 0

Graphs:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Binomial Disrtibution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Poisson Distribution

X

Prob

abili

ty

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Normal Distribution

X

Prob

abili

ty

Probability distribution for p=0.1

n=5,

X Binomial distribution Poisson distribution Normal distribution0 0.59049 0.60653066 0.4319814361 0.32805 0.30326533 0.4319814362 0.0729 0.075816332 0.0665840083 0.0081 0.012636055 0.0014306844 0.00045 0.001579507 3.87087E-065 0.00001 0.000157951 1.24E-09

Graphs:

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Binomial Distribution

X

Prob

abili

ty

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Poisson Distribution

X

Prob

abili

ty

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

Normal Distribution

X

Prob

abili

ty

n=10,

X Binomial distribution Poisson distribution Normal distribution0 0.348678 0.367879441 0.2421575771 0.38742 0.367879441 0.4018385472 0.19371 0.183939721 0.2421575773 0.057396 0.06131324 0.0527191524 0.01116 0.01532831 0.0040915725 0.001488 0.003065662 0.0001113756 0.000138 0.000510944 1.04735E-067 8.75E-06 7.2992E-05 3.36E-098 3.65E-07 9.12399E-06 3.65E-129 9E-09 1.01378E-06 1.33E-1510 1E-10 1.01378E-07 0

Graphs:

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Binomial Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Poisson Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Normal Distribution

X

Prob

abili

ty

n=20,

X Binomial distribution Poisson distribution Normal distribution0 0.121577 0.135335283 0.1005725291 0.27017 0.270670566 0.2229178192 0.28518 0.270670566 0.2906118853 0.19012 0.180447044 0.2229178194 0.089779 0.090223522 0.1005725295 0.031921 0.036089409 0.0266599766 0.008867 0.012029803 0.0041456197 0.00197 0.003437087 0.0003774098 0.000356 0.000859272 2.00723E-059 5.27E-05 0.000190949 6.22309E-0710 6.44E-06 3.81899E-05 1.12E-0811 6.51E-07 6.94361E-06 1.18E-1012 5.42E-08 1.15727E-06 7.14E-1313 3.71E-09 1.78041E-07 2.55E-1514 2.06E-10 2.54345E-08 0

15 9.15E-12 3.39126E-09 016 3.18E-13 4.23908E-10 017 8.31E-15 4.98715E-11 018 1.54E-16 5.54128E-12 019 1.8E-18 5.83292E-13 020 1E-20 5.83292E-14 0

Graphs:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.05

0.1

0.15

0.2

0.25

0.3

Binomial Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.05

0.1

0.15

0.2

0.25

0.3

Poisson Distribution

X

Prob

abili

ty

0 2 4 6 8 10 12 14 16 18 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Normal Distribution

X

Prob

abili

ty

Probability distribution for p=0.5

n=5

X Binomial distribution Poisson distribution Normal distribution0 0.03125 0.082085 0.033174

1 0.15625 0.205212 0.148728

2 0.3125 0.256516 0.314453

3 0.3125 0.213763 0.314453

4 0.15625 0.133602 0.148728

5 0.03125 0.066801 0.033174

Graphs:

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Binomial Distribution

X

Prob

abili

ty

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

Poisson Distribution

X

Prob

abili

ty

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Normal Distribution

X

Prob

abili

ty

n=10,

X Binomial distribution Poisson distribution Normal distribution0 0.000977 0.006738 0.001961

1 0.009766 0.03369 0.011215

2 0.043945 0.084224 0.043495

3 0.117188 0.140374 0.114468

4 0.205078 0.175467 0.204524

5 0.246094 0.175467 0.24817

6 0.205078 0.146223 0.204524

7 0.117188 0.104445 0.114468

8 0.043945 0.065278 0.043495

9 0.009766 0.036266 0.011215

10 0.000977 0.018133 0.001961

Graphs:

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

Binomial Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Poisson Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

Normal Distribution

X

Prob

abili

ty

n=20,

X Binomial distribution Poisson distribution Normal distribution0 9.53674E-07 4.53999E-05 9.43073E-061 1.90735E-05 0.000453999 6.1206E-052 0.000181198 0.002269996 0.000326153 0.001087189 0.007566655 0.0014271024 0.004620552 0.018916637 0.0051279315 0.014785767 0.037833275 0.0151325246 0.036964417 0.063055458 0.0366767627 0.073928833 0.090079226 0.0730138058 0.120134354 0.112599032 0.1193912399 0.160179138 0.125110036 0.1603641610 0.176197052 0.125110036 0.17693672611 0.160179138 0.113736396 0.1603641612 0.120134354 0.09478033 0.11939123913 0.073928833 0.072907946 0.07301380514 0.036964417 0.052077104 0.036676762

15 0.014785767 0.03471807 0.01513252416 0.004620552 0.021698794 0.00512793117 0.001087189 0.012763996 0.00142710218 0.000181198 0.007091109 0.0003261519 1.90735E-05 0.003732163 6.1206E-0520 9.53674E-07 0.001866081 9.43073E-06

Graphs:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Binomial Distribution

X

Prob

abili

ty

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Poisson Distribution

X

Prob

abili

ty

0 2 4 6 8 10 12 14 16 18 200.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

1.80E-01

2.00E-01

Normal Distribution

X

Prob

abili

ty

Figure 1

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Comparison of Normal Distributions for p=0.05

n=5n=10n=20

X

P(X

=x)

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Comparison of Normal Distributions for p=0.1

n=5

n=10

n=20

X

P(X

=x)

Figure 2

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Comparison of Normal Distributions for p=0.5

n=5n=10n=20

X

P(X

=x)

Figure 3

Conclusion

When the sample size, n and the successful probability, p is not sufficient large, the probability distributions obtained are positively-skewed. However, as n and p increases, the graphs of probabilities become more normal and symmetrical. Thus, theoretically, it can be concluded that the graphs for normal distribution will be a perfect bell-shaped if the sample size, n is approaching infinity.

For p=0.05,the shape of the graphs is not symmetrical and skewed to the right.The shape of graph of the Poisson distribution and the binomial distribution are similar while the graph of normal distribution skewed to the right significantly although n increases. Thus,normal approximation is not suitable in these case as the skew of the distribution is too great.

From Figure 2,the shape of normal distribution for n=5 still tends to skew to the right,as n=10 and n=20, the shape of graph for normal distribution less skew to the right.This shows that when n increases, the graph become more normal.This also can be seen from the graphs of binomial distribution and poisson distribution.

From Figure 3, when p=0.5,all the shape of graph for normal distribution shown are fundamentally normal and symmetrical for all n values (n=5,n=10 and n=20).For n=20 which is sufficiently large, the graph shown has an almost bell-shape curve. This have shown that when the value of p and n is sufficiently large, normal approximation is sensible for calculating probability since the skew of the distribution is less. The binomial distribution graph and poisson distribution graph also reveal the similar pattern. As n increase the probability distribution will show a more normal and symmetrical graphs. The graphs will theoretically become a prefect bell-shaped if the sample size, n is approaching infinity.