A binomial random variable X is defined as the number of successes achieved in the n trials of a Bernoulli process. Describe an event in your life that fits the properties of a Bernoulli process, being sure to explain how each property is met by your event. Finally, state the number of trials and the number of successes for your event. Be specific.
Recommended materials
Biscarri, W., Zhao, S.D., & Brunner, R.J. (2018). A simple and fast method for computing the Poisson binomial distribution function Links to an external site.. Computational Statistics and Data Analysis, 122, 92-100.
Saurabh, P., & Divesh, S. (2017). Harmonic starlikeness and convexity of integral operators generated by Poisson distribution series. Mathematica Moravica, 21(1), 51-60. Retrieved from http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2017/1450-59321701051P.pdfLinks to an external site.
References
(2019). Introductory statistics. Houston, TX: OpenStax College. CC BY-SA. Retrieved from https://cnx.org/contents/MBiUQmmY@23.21:kcV4GRqc@17Links to an external site.
Discrete Probability Distributions
A random variable can be:
Discrete – it is countable and finite (has a limit).
Continuous – it is infinite (no limit) and measurable.
How do we distinguish between the two of them?
Use the key question – can we measure it?
if yes – then we have a continuous random variable (keywords: length, width, volume, height, weight, scores, salary,…., are all measurable).
if no – then we have a discrete random variable (keywords: number of books in a library, phone numbers, social security numbers, floor and room numbers in a hotel, number of cars in a parking lot has a limit/capacity,…, are all countable).
There are two conditions that must be met in order to have a discrete probability distribution:
Each probability is between 0 and 1 (Probabilities cannot be negative or greater than 100%).
The sum of all probabilities is 1 (or 100%).
If one or both conditions are not met then we don’t have a discrete probability distribution.
Binomial Probability Distributions
In a binomial probability, we can only have 2 outcomes: success and failure.
In a binomial experiment we need to know:
n = number of independent trials (surgeries, for instance, must be done separately, this is why we use the word “independent”.)
p = probability of success
q = 1-p probability of failure
x = number of successes only (or successful trials only).
Class example:
A surgeon has scheduled 8 patients for a new eye procedure. Since it is new (thus experimental, with many unknowns), this surgery has a probability of success of only 63%. Is this a binomial experiment?
Let’s see if we can find all 4 parameters:
n = 8
p = 0.63 (change the percent into a decimal number as we don’t do operations with percents)
q = 1 – p = 1 – 0.63 = 0.37
x = 0,1,2,3,4,5,6,7,8.
Conclusion: In this case, we can have the following possible outcomes/situations (x means the following):
a. all 8 surgeries are successful or
b. some surgeries are successful (let’s say 2 or 5 of them) or
c. NONE of them is successful – this is why zero is very important in this case.
This is a probability experiment since we can identify all 4 parameters of a binomial experiment.
The easiest formulas for the mean, median, and standard deviation are those for a Binomial Probability Distribution:
Mean = n*p
Variance = n*p*q and
Standard dev. = sq. root of n*p*q
Very Important notes:
If probability of success, p is a multiple of 5 (0.15, 0.30, 0.65, and so on) we use the binomial table – see table 4 in the attached doc.
If probability of success, p is NOT a multiple of 5 (0.17, 0.34, 0.63, and so on), then we use the binomial probability function (formula).
