Normal, Exponential, And Poisson Distribution Contrast

Normal, Exponential, and Poisson distribution contrast
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Normal Distribution
A normal distribution refers to a function representation of numerous random variables in a symmetrical bell-shaped curve. This distribution is commonly used because of the predictive symmetrical probability graph that peaks at the mean (μ) and is easy to use.
In a situation where the mean changes, the graph moves along the x-axis with respect to the new mean (μ) since the curve is symmetrical with respect to the mean (μ) of the distribution. On the other hand, when the standard deviation (σ) of the distribution changes, the probability range shrinks in the case of small S.D (σ) and spreads in the case of a large S.D (σ).
The normal distribution is defined by the below equation:
Y = {12π} * e-(x-μ)22σ2
X = random variable
π ≈ 3.14159
e ≈ 2.71828
μ = mean
σ = standard deviation
It is easy to check a desired factor or probability under normal distribution using the normal distribution tables that converts the distribution into a standard normal distribution. When using the table, the mean (μ) is set at 0 while the standard deviation (σ) is set at 1 (Brereton, 2014). A new variable, Z, is established and is part of the normal distribution table. The formula for calculating Z is as below:
Z = x- μσ
Z = number of standard deviations from mean
X = random variable
μ = mean
σ = standard deviation
Under the normal distribution, researchers established a rule that is found to be true, called the 68-95-99.7 rule that states:
* Approximately 68% of the values under the area of the curve are between ±1 standard deviation of the mean.
* Approximately 95% of the values under the area of the curve are between ±2 standard deviation of the mean.
* Approximately 99.7% (almost all) of the values under the area of the curve are between ±3 standard deviation of the mean.
Example
The heights of students in a class are normally distributed with a mean of 150cm and a standard deviation of 6cm.
* Find the probability that a randomly selected student has a height greater than 160cm
* Find the probability that a randomly selected student has a height less than 140cm
Solution
* P(X>160)
μ = 150
Z = x- μσ = 160 - 1506 = 1.67
P(x>160) = P(z>1.67) = 0.0478
* P(X<140)
μ = 150
Z = x- μσ = 140 - 1506 = - 1.67
P(x<140) = P(z<-1.67) = 0.0478
Exponential distribution
When focusing on the exponential distribution, the natural logarithm e is very important as it is one of the bas...