Probability distributions, Biology tutorial

Introduction:

Define: A probability distribution is an equation or a table or that links each and every outcome of a statistical experiment by its probability of occurrence.

A distribution is basically a scatter of associated values, like the assortment of weights in a group of cattle. A frequency distribution exhibits how many times given values in a range of values take place. A Probability distribution is much similar as it exhibits us how probable given random variable values in a range of such values are.

An illustration will make apparent the relationship among random variables and probability distributions. Assume that you flip a coin two times. This simple statistical experiment can encompass four possible outcomes: HH, HT, TH and TT. Now, assume that the variable X symbolizes the number of Heads which result from this experiment. The variable X can take on the values 0, 1 or 2. In this instance, X is a random variable; as its value is found out by the outcome of the statistical experiment.

The table beneath associates each outcome with the probability, is an illustration of a probability distribution.

Number of heads        Probability

0                                 0.25

1                                 0.50

2                                 0.25

The table above symbolizes the probability distribution of the random variable X.

The Normal Distribution:

This is the most significant distribution in statistics. It is as well termed as the Gaussian distribution termed after Gauss, a German astronomer who exhibited its utilization in statistics. The normal distribution is stated by merely two statistics, the mean and the standard deviation. Normal distribution is mainly concerned with outcomes obtained by taking measurements on continuous random variable (that is, the quantified value of a random event) similar to weight, yield and so on. The normal distribution is a specific pattern of variation of numbers about the mean. It is symmetrical (therefore we state the standard deviation as ±) and the frequency of individual numbers falls off uniformly away from the mean in both directions. In terms of human height, gradually bigger and smaller people than the average take place symmetrically by reducing frequency correspondingly giants or dwarfs. What is significant regarding this distribution is not merely that this type of natural variation often takes place, however as well that it is the distribution that comes with the best statistical reference for data analysis and testing of hypotheses. It so occurs that the curve given by this probabilities distribution estimate very closely to a Mathematical curve.  This curve is termed as the Normal curve.

In examining for normality, it is significant to know whether an experimental data is an estimated fit to a normal distribution. This is simply checked with big samples. There must be roughly equivalent numbers of observations on either side of the mean. Things are more complicated when we encompass only a few samples. In experiments, it is not rare to have no more than three data per treatment. Though, even here we can get clues. When the distribution is normal, there must be no relationship among the magnitude of the mean and its standard deviation. 

Properties of a Normal Curve:

a) This is a Unimodal symmetrical curve.

b) The mean, mode and median all overlap, thus dividing the curve into two equivalent parts.

c) Most of the items on the curve are clustered about the mean.

d) No kurtosis or skewness in the curve.

e) The area under the curve is proportional to the observation relating with the part.

Standardizing the normal curve:

Any value of the observation X on the baseline of a normal curve can be standardized as the number of standard deviation units, the observation is absent from the population mean, μ.  This is termed as a z-score. To convert x into z the formula is given by:

z = (X - μ)/ σ

When the population mean µ is bigger than the sample mean x, the z is negative. However when the sample size is greater than around 30 observations, the sample mean (x) and standard deviation (s) are considered to be excellent estimates of μ and σ, and z is represented by:

z = (X - μ)/S

When the calculated value of z is bigger than 1.96 (that is, P < 0.05 or 95% confidence coefficient) then this is considered as unlikely or statistically significant.

Poisson Distribution:

A Poisson distribution is a discrete probability distribution which is helpful when n is bigger and p is small and if the independent variables take place over a period of time.  It can be employed if a density of items is distributed over a specific area or volume, like the number of plants growing per acre. It can as well be employed to discover whether organisms are randomly distributed. For instance, in ecological studies, Poisson distribution is employed to explain the spread of organisms such as trees, insects, snails and so on.

a) Divide the bigger area into small squares of equivalent size.

b) Count the specific animal or plant species beneath study in each square.

c) You can as well arbitrarily choose a number of squares, when the area is too large.

The probability of X occurrences in an interval of time, volume, area and so on for a variable where λ (lambda) is the mean number of occurrences per unit (time, volume, area and so on) is given by: 

P (x, λ) = (l λx)/x!

where x = 0, 1, 2, 3,........

e = constant, approximately equivalent to 2.7183

Binomial Distribution:

A binomial experiment is a probability experiment which satisfies the given four necessities:

1) Each and every trial can encompass just two outcomes or outcomes which can be decreased to two outcomes that is, such outcomes can either be failure or success. No two events can take place concurrently.

2) There should be a fixed number of trails.

3) The outcomes of each and every trial should be independent of one other.  

4) The probability of a success should remain similar for each and every trial.

A binomial distribution is a special probability distribution which explains the distribution of probabilities if there are just two possible outcomes for each and every trial of an experiment.

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