Probability Distributions

Binomial Distribution

Discovered by Swiss Mathematician James Bernoulli hence it is also known as Bernoulli’s distribution. There are many trials that have only two outcomes. Such single trials are called binomial or Bernoulli’s trials.

The binomial distribution can be defined as the probability distribution of discrete variable where there are only two possible out comes for each trial of an experiment. In this case both of these outcomes are important.

Assumptions of binomial distribution:

• Each trial should have only two possible outcome “success” or “failure”.
• The probability of success or failure should remain constant from trial to trial.
• The events should be independent of each other.
• The trials should be repeated for a fixed number of times, i.e., ‘n’ should be finite.

The sample space in binomial distribution consists of only two elements. If we denote the probability of success as ‘p’ and that of failure as ‘q’, then the probability space will be:

In two trials: { SS , SF, FF} { p², 2pq, q²}

In three trials: {SSS, SSF, SFF, FFF} { p³, 3p²q, 3pq², q³ }

Binomial expansion is accomplished by the binomial term (p + q)ⁿ

where, n = sample size, p = probability of occurrence of first class, and q = probability of occurrence of the second class.

Extension of samples from 1 to 4 will be:

1. (p + q)¹ = p + q

2. (p + q)² = p² + 2pq + q²

3. (p + q)³ = p³ + 3p²q + 3pq² + q³

4.(p + q)⁴ = p⁴ + 4p³q + 6p²q² + 4pq³ + q⁴

The coefficients (the numbers before the powers of p and q) express the number of ways a particular outcome is obtained and are known as binomial coefficients. Coefficients of the expanded term of the binomial expression by Pascal’s Triangle

r Coefficients

0 1

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

5 1 5 10 10 5 1

6 1 6 15 20 15 6 1

7 1 7 21 35 35 21 7 1

8 1 8 28 56 70 56 28 8 1

Properties of binomial distribution:

• The sum of probabilities of 0,1, 2 , …..…, n successes is unity: S p( r) = 1.
• The mean of binomial distribution is np and its variance is npq.
• The quantity n and p are the parameters of the binomial distribution and they determine the shape and location of distribution.
• For ‘n’ trials, a binomial distribution consists of (n + 1) terms, the successive binomial coefficients being ⁿC₀, ⁿC₁, ⁿC₂, ……….. ⁿC_{(n –1)} and ⁿC_n.
• If p = q = ½, the distribution is symmetrical and the expected frequencies on either side of the central value would be identical.
• A binomial distribution is positively skewed if p < ½ and it is negatively skewed if p > ½.
• The mean of the distribution increases as ‘n’ increases with ‘p’ held constant.
• The variance of a binomial distribution is always less than its mean and both p and q are less than one but not equal to zero.

Poisson Distribution

Discovered by French Mathematician S. D. Poisson.

The Poisson distribution is a discrete frequency distribution of the number of times a rare event occurs.

However, in contrast to the binomial distribution, the number of times that an event does not occur is infinitely large.

Examples:

1. Birth of a calf having congenital defect
2. Presence of an abnormal cyst in an internal organ
3. Persons killed by kicking of horses
4. Number of people dying in road accidents etc.

Assumptions of Poisson Distribution

1. The happening or non-happening of any event should not affect those of any other events, i.e. events should be independent.
2. The probability of happening of more than one event in a very small interval should be negligible.
3. Poisson distribution is a limiting form of binomial distribution.
4. The Poisson distribution may be used to approximate the binomial distribution when n > 50 and p < 0.1

Properties of Poisson Distribution

1. The mean of this distribution in np.
2. The variance of Poisson distribution is also equal to its mean, i.e. np.
3. The parametric mean m is the constant of the Poisson distribution. This determines the shape of distribution.

Uses of Poisson Distribution

• In biology to count number of bacteria in a unit cell.
• In statistical quality control to count number of defects with a product.
• In insurance problems to determine the number of casualties.
• In printing/typing business to know the number of typographical mistakes per page.
• In birth and death registration offices to determine the number of deaths due to a rare disease in a particular locality.
• To count number of road accidents, number of suicides, and many more such rare cases in day-to-day life.

Normal Distribution

Properties of the Normal Distribution

1. The normal probability density function is:

2. There are two parameters of a normal distribution; m, the parametric mean and s, the parametric standard deviation, which determine the location and the shape of the distribution.

3. Theoretically a normal distribution extends from negative infinity to positive infinity along the axis of the variable.

4. The curve is bell shaped, unimodal (has only one peak) and symmetrical around the mean.

5. The mean, median and mode of a normally distributed curve are all at the same point (mean = median = mode).

6. The tails of the curve taper towards the base line on both the sides asymptotically, i.e., they do not touch the base line.

7. The percentage of items in a normal distribution:

mu ± sd contains 68.26 % of the items

mu ± 2 sd contains 95.45 % of the items

mu ± 3 sd contains 99.73 % of the items

8. The coefficient of skewness in a normal curve is zero while the coefficient of kurtosis is 3.

9. A normal curve with zero mean and unit variance is referred to as the standard normal curve and then the variate is known as standard normal variate or standard normal deviate. We can transform a normal curve into a standard normal curve by Z-transformation as:

Applications of Normal Distribution:

1. It forms the basis of tests of significance.
2. It is widely used in planning of experiments.
3. It is highly useful in statistical quality control in industries for setting up of the control limits.
4. It is very much used in finding out the probabilities of various sample results.
5. It is most often used in the large sampling theory to find the estimates of the parameters and confidence limits from statistic.
6. It also serves as an approximation to the binomial and Poisson distributions under certain conditions.