Probability Density Functions
Order ID 
53563633773 
Type 
Essay 
Writer Level 
Masters 
Style 
APA 
Sources/References 
4 
Perfect Number of Pages to Order 
510 Pages 
Description/Paper Instructions
Probability Density Functions (PDFs) are functions of mathematics that define an ongoing randomized variable’s probability distribution. They provide a method to calculate the probability of a random variable taking on distinct values within a particular range.
A PDF is a nonnegative function that integrates to 1 throughout its entire domain, represented by f(x). This signifies that the PDF’s area under the curve equals 1. The Portable Document Format (PDF) is defined for all random variable x values within a particular range of values.
The normal distribution, often known as the Gaussian distribution or the bell curve, is a popular example of a PDF. A normal distribution’s PDF is given by:
2f(x)=2πσ21e−2σ2(x−μ)2
Here, indicates the distribution’s mean, and is the standard deviation. This PDF describes the probability that a random number x will take on distinct values surrounding the mean. The higher the curve’s peak, the greater the probability of values close the mean are, while the curve’s tails reflect a greater likelihood of extreme values.
The uniform distribution is another important PDF which is frequently used when every potential outcome in a range were equally probable.
The PDF for an even distribution across the range [a, b] can be obtained by: ( )=1,if 0,elsef(x)=ba1,0,if axbotherwiseIn this situation, all values inside the range [a, b] have an identical probability weight, but values beyond this range have a probability density of 0.
Another prominent PDF is the exponential distribution, which is frequently employed to simulate the duration between occurrences in a Poisson process. Its PDF is obtained by: ( )=,if 00,otherwisef(x)=ex,0,if x0,otherwise, the rate parameter, in this case, specifies the pace at which occurrences occur.PDFs are essential in probability and statistics.
This is because they allow us to calculate probabilities associated with specific outcomes or ranges of values for continuous random variables. To find the probability that a continuous random variable falls within a certain interval, we can integrate the PDF over that interval.
In the end, Probability Density Functions are mathematical instruments that are used to represent the distribution of probabilities of continuous random variables. They give significant insight into the likelihood of various events and are used in a variety of industries, including statistics, engineering, physics, and finance. PDFs of various forms, including normal, uniform, and exponentially payments, are utilized to explain many realworld events and are required for making probabilistic forecasts and judgments.
RUBRIC
QUALITY OF RESPONSE 
NO RESPONSE 
POOR / UNSATISFACTORY 
SATISFACTORY 
GOOD 
EXCELLENT 
Content (worth a maximum of 50% of the total points) 
Zero points: Student failed to submit the final paper. 
20 points out of 50: The essay illustrates poor understanding of the relevant material by failing to address or incorrectly addressing the relevant content; failing to identify or inaccurately explaining/defining key concepts/ideas; ignoring or incorrectly explaining key points/claims and the reasoning behind them; and/or incorrectly or inappropriately using terminology; and elements of the response are lacking. 
30 points out of 50: The essay illustrates a rudimentary understanding of the relevant material by mentioning but not full explaining the relevant content; identifying some of the key concepts/ideas though failing to fully or accurately explain many of them; using terminology, though sometimes inaccurately or inappropriately; and/or incorporating some key claims/points but failing to explain the reasoning behind them or doing so inaccurately. Elements of the required response may also be lacking. 
40 points out of 50: The essay illustrates solid understanding of the relevant material by correctly addressing most of the relevant content; identifying and explaining most of the key concepts/ideas; using correct terminology; explaining the reasoning behind most of the key points/claims; and/or where necessary or useful, substantiating some points with accurate examples. The answer is complete. 
50 points: The essay illustrates exemplary understanding of the relevant material by thoroughly and correctly addressing the relevant content; identifying and explaining all of the key concepts/ideas; using correct terminology explaining the reasoning behind key points/claims and substantiating, as necessary/useful, points with several accurate and illuminating examples. No aspects of the required answer are missing. 
Use of Sources (worth a maximum of 20% of the total points). 
Zero points: Student failed to include citations and/or references. Or the student failed to submit a final paper. 
5 out 20 points: Sources are seldom cited to support statements and/or format of citations are not recognizable as APA 6^{th} Edition format. There are major errors in the formation of the references and citations. And/or there is a major reliance on highly questionable. The Student fails to provide an adequate synthesis of research collected for the paper. 
10 out 20 points: References to scholarly sources are occasionally given; many statements seem unsubstantiated. Frequent errors in APA 6^{th} Edition format, leaving the reader confused about the source of the information. There are significant errors of the formation in the references and citations. And/or there is a significant use of highly questionable sources. 
15 out 20 points: Credible Scholarly sources are used effectively support claims and are, for the most part, clear and fairly represented. APA 6^{th} Edition is used with only a few minor errors. There are minor errors in reference and/or citations. And/or there is some use of questionable sources. 
20 points: Credible scholarly sources are used to give compelling evidence to support claims and are clearly and fairly represented. APA 6^{th} Edition format is used accurately and consistently. The student uses above the maximum required references in the development of the assignment. 
Grammar (worth maximum of 20% of total points) 
Zero points: Student failed to submit the final paper. 
5 points out of 20: The paper does not communicate ideas/points clearly due to inappropriate use of terminology and vague language; thoughts and sentences are disjointed or incomprehensible; organization lacking; and/or numerous grammatical, spelling/punctuation errors 
10 points out 20: The paper is often unclear and difficult to follow due to some inappropriate terminology and/or vague language; ideas may be fragmented, wandering and/or repetitive; poor organization; and/or some grammatical, spelling, punctuation errors 
15 points out of 20: The paper is mostly clear as a result of appropriate use of terminology and minimal vagueness; no tangents and no repetition; fairly good organization; almost perfect grammar, spelling, punctuation, and word usage. 
20 points: The paper is clear, concise, and a pleasure to read as a result of appropriate and precise use of terminology; total coherence of thoughts and presentation and logical organization; and the essay is error free. 
Structure of the Paper (worth 10% of total points) 
Zero points: Student failed to submit the final paper. 
3 points out of 10: Student needs to develop better formatting skills. The paper omits significant structural elements required for and APA 6^{th} edition paper. Formatting of the paper has major flaws. The paper does not conform to APA 6^{th} edition requirements whatsoever. 
5 points out of 10: Appearance of final paper demonstrates the student’s limited ability to format the paper. There are significant errors in formatting and/or the total omission of major components of an APA 6^{th} edition paper. They can include the omission of the cover page, abstract, and page numbers. Additionally the page has major formatting issues with spacing or paragraph formation. Font size might not conform to size requirements. The student also significantly writes too large or too short of and paper 
7 points out of 10: Research paper presents an aboveaverage use of formatting skills. The paper has slight errors within the paper. This can include small errors or omissions with the cover page, abstract, page number, and headers. There could be also slight formatting issues with the document spacing or the font Additionally the paper might slightly exceed or undershoot the specific number of required written pages for the assignment. 
10 points: Student provides a highcaliber, formatted paper. This includes an APA 6^{th} edition cover page, abstract, page number, headers and is double spaced in 12’ Times Roman Font. Additionally, the paper conforms to the specific number of required written pages and neither goes over or under the specified length of the paper. 


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