Weekly Summary 4.1

19 May No Comments

Weekly Summary 4.1

Course Title: CBSC520 Data Analytics

University of the Potomac

Abstract

In this paper we discussed about what we have learned in this week and following is a summary of what we understood. This week class depicts the topics of what is Normal distribution, Binomial distribution, Exponential distribution and Poisson distribution?

The normal distribution is the most important probability distribution in statisticsbecause it fits many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. It is also known as the Gaussian distribution and the bell curve.

The normal distribution is a probability function that describes how the values of a variable are distributed. It is a symmetric distribution where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions. Extreme values in both tails of the distribution are similarly unlikely.

https://statisticsbyjim.com/basics/normal-distribution/

Example of Normally Distributed Data: Heights

Height data are normally distributed. The distribution in this example fits real data that I collected from 14-year-old girls during a study.

As you can see, the distribution of heights follows the typical pattern for all normal distributions. Most girls are close to the average (1.512 meters). Small differences between an individual’s height and the mean occur more frequently than substantial deviations from the mean. The standard deviation is 0.0741m, which indicates the typical distance that individual girls tend to fall from mean height.

The distribution is symmetric. The number of girls shorter than average equals the number of girls taller than average. In both tails of the distribution, extremely short girls occur as infrequently as extremely tall girls.

Normal Approximation Of The Binomial Distribution

The binomial distribution is a discrete probability distribution that is used to obtain the probability of observing exactly k number of successes in a sequence of n trials, with the probability of success for all single trials of p.

The shape of the binomial distribution is dependent on the values of n and p, as illustrated in the following diagrams:

The normal distribution is a continuous probability distribution whose shape is dependent on the mean, μ, and variance, σ.

 Since a binomial variate, B(n,p), is a sum of n independent, identically distributed Bernoulli variables with parameterp, it follows that by the central limit theorem it can be approximated by the normal distribution with mean n p and variance n p (1−p), provided that both n p >5 and n (1−p) > 5.

https://www.maplesoft.com/support/help/maple/view.aspx?path=MathApps%2FNormalApproximationOfBinomialDistribution

What is a Binomial Distribution?

binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes (the prefix “bi” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.

A Binomial Distribution shows either (S)uccess or (F)ailure.
The first variable in the binomial formula, n, stands for the number of times the experiment runs. The second variable, p, represents the probability of one specific outcome. For example, let’s suppose you wanted to know the probability of getting a 1 on a die roll. if you were to roll a die 20 times, the probability of rolling a one on any throw is 1/6. Roll twenty times and you have a binomial distribution of (n=20, p=1/6). SUCCESS would be “roll a one” and FAILURE would be “roll anything else.” If the outcome in question was the probability of the die landing on an even number, the binomial distribution would then become (n=20, p=1/2). That’s because your probability of throwing an even number is one half.

Real Life Examples

Many instances of binomial distributions can be found in real life. For example, if a new drug is introduced to cure a disease, it either cures the disease (it’s successful) or it doesn’t cure the disease (it’s a failure). If you purchase a lottery ticket, you’re either going to win money, or you aren’t. Basically, anything you can think of that can only be a success or a failure can be represented by a binomial distribution.

The binomial distribution formula is:

b(x; n, P) = nCx * Px * (1 – P)n – x

Where:
b = binomial probability
x = total number of “successes” (pass or fail, heads or tails etc.)
P = probability of a success on an individual trial
n = number of trials

Note: The binomial distribution formula can also be written in a slightly different way, because nCx = n!/x!(n-x)! (this binomial distribution formula uses factorials (What is a factorial?). “q” in this formula is just the probability of failure (subtract your probability of success from 1).

https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/binomial-theorem/binomial-distribution-formula/

If X has a binomial distribution with n trials and probability of success p on each trial, then:

The mean of X is

The variance of X is

The standard deviation of X is

For example, suppose you flip a fair coin 100 times and let X be the number of heads; then X has a binomial distribution with = 100 and p = 0.50. Its mean is

heads (which makes sense, because if you flip a coin 100 times, you would expect to get 50 heads). The variance of X is

which is in square units (so you can’t interpret it); and the standard deviation is the square root of the variance, which is 5. That means when you flip a coin 100 times, and do that over and over, the average number of heads you’ll get is 50, and you can expect that to vary by about 5 heads on average.

Poisson Distribution

Basic Concepts

Definition 1: The Poisson distribution has a probability distribution function (pdf) given by

The parameter μ is often replaced by λ. A chart of the pdf of the Poisson distribution for λ= 3 is shown in Figure 1.

Figure 1 – Poisson Distribution

Observation: Some key statistical properties of the Poisson distribution are:

Mean = µ

Variance = µ

Skewness = 1 /

Kurtosis = 1/µ

Excel Function: Excel provides the following function for the Poisson distribution:

POISSON(x, μ, cum) where μ = the mean of the distribution and cum takes the values TRUE and FALSE

POISSON(x, μ, FALSE) = probability density function value f(x) at the value x for the Poisson distribution with mean μ.

POISSON(x, μ, TRUE) = cumulative probability distribution function F(x) at the value x for the Poisson distribution with mean μ.

Excel 2010/2013/2016 provide the additional function POISSON.DIST which is equivalent to POISSON.

http://www.real-statistics.com/binomial-and-related-distributions/poisson-distribution/

Exponential Distribution Formula

The exponential function is a special type where the input variable works as the exponent. A function f(x) = bx + c or function f(x) = a, both are the exponential functions. It is used everywhere, if we talk about the C programming language then the exponential function is defined as the e raised to the power x. Here, x could be any real number. The syntax for exponential functions in C programming is given as –

https://www.andlearning.org/exponential-formula/

Albright, S. & Winston, W. (2017). Business Analytics – Data Analysis and Decision Making: Cengage Learning,




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