Binomial distribution
When do we use the binomial distribution?
The binomial distribution is very useful when there are very many trials conducted in an experiment, say n, and we want to get the probability of finding success where each number of successes has a given probability of occurrence. The probability of success is denoted by p while that of failure is the complement, i.e. (1-p) (Collani & Dräger, 2011). The binomial distribution gives some of the probabilities that are associated with repeated Bernoulli independent trials.
Characteristics of binomial distribution
For a binomial experiment, an action is repeated several definite number of times
The condition for the repetition of the given experiments must be equal and identical; this is to say that there is no bias in the binomial experiments
Each of the trial in the binomial experiment must not be dependent of other trials, they should be independent. For the binomial experiment there must only be two possible set of outcomes, they can either be failure or success, head and tail, bad and good etc. The possible number of the successful experiments or trials must be less than the total number of trials.
The mean of the binomial distribution is denoted by
μ=np
Where n is the total number of trials, and p is the probability of success in the experiment
The standard deviation of the binomial distribution is given by
σ=npq
Where n is the total number of trials, p is the probability of success and q is the probability of failure in the experiment
When the values of both np and nq are higher than five then the histogram of the binomial closely resembles that of the normal distribution. The normal distribution has its mean, median and the mode all equal and is symmetric. That is how the binomial distribution will resemble when the values of both np and nq are higher than five.
When the probability of success is equal to the probability of failure then the curve of the distribution is of symmetry about the distribution mean. This is to mean that the probability of having a success is equal to that of having a failure.
For any binomial experiment with the probability of success equal to the probability of failure, there are only 11 trials that are needed to suit the condition of 0.5*11 greater than 5
When the value of the probability of success gets away from 0.5, there are very many numbers of trials that are required to bring the symmetry in the distribution curve. Take for example if there the probability of success is 0.1, this means that the probability of failure is 0.9. From this kind of experiment there at least 51 number of trial that are needed so that the binomial distribution curve can be identical to the normal distribution curve.
The binomial probabilities that are not on the binomial table can be computed by the use of the curve of the normal distribution in the place of the histogram of the binomial distribution. Take for example when the number of trials exceeds 20 in number.
The binomial probabilities that appear to be very tedious to compute using the formula of binomial can be computed by the use of the curve of the normal distribution in the place of the histogram of the binomial distribution.
Normal distribution
When do we use the normal distribution?
The normal distribution is used in the case where the sample data appears to have varied characteristics; this is when the data is spread over a larger scale that appears to be normal. This is because as the sample size increases, the non normal distributions tend to approach the normal distribution (Bryc, 2015).
Characteristics of normal distribution
The normal distribution is bell shaped and symmetric this is to say that the rights side is the mirror image of the other side
The normal distribution has only one peak or node
The tails of the curve of the normal distribution are asymptotic, this is to say that they are usually close to the X axis but they don’t touch on the axis
It is continuous for the values of x in the range of negative infinity to infinity
The fifty percent of the normal distribution data lies above its mean and the other 50% lie below the mean. Therefore the mean is at the mark of 50%
The two parameters that determine its shape are the mean and standard deviation
The formula is given by
fx;μ,σ2=12πσ2e-(x-μ)22σ2
This notation N(μ.σ2) means that the sample is normally distributed with a meanμ and variance of σ2. If it is written that X~N(μ.σ2), that mean that x is normally distributed N(μ.σ2)
About the two thirds of all the cases lie within 1 standard deviation in the mean of the distribution i.e.
All the normal distribution curves have asymmetry about the distribution mean
P (μ –σ ≤ X ≤μ + σ) =0 .6826.
About the ninety five percent of all the cases are within two standard deviation in the mean i.e.
P (μ –2σ ≤ X ≤μ + 2σ) =0.9544
The total area under the normal distribution curve is equal to 1
All the normal distribution curves have positive values for all y. this is to say that f(x) > 0 for all the possible values of x
The maximization of the height of any normal distribution curve is at when x=μ
The normal distribution curves are always denser in its centre and are less denser in its tail region
The mean, the median and the mode of the normal distribution are all equal in values.
The normal distribution is also called the Gaussian distribution.
The normal distribution can be skewed to the right; this is to say that the tail is thin on the right hand side of the normal curve. The distribution can also be negatively skewed; this is to say that the tail is thin on the left side.
For the positive skewness, the mode and the median are behind the mark of the mean, this is to say that the mode and the median are less than the mean.
For negative skewness, the mode and that median are on the right side of the mean. Meaning that the mode and the median are all greater than the distribution mean.
References
Collani, E. ., & Dräger, K. (2011). Binomial distribution handbook for scientists and engineers. Boston [u.a.: Birkhäuser.
Bryc, W. (2015). The normal distribution: Characterizations with applications. New York: Springer-Verlag.
