Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips

Prepare for your exams

Study with the several resources on Docsity

Earn points to download

Earn points by helping other students or get them with a premium plan

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

An introduction to probability distributions, including the difference between continuous and discrete variables, and examples of common continuous probability density functions (pdfs) such as uniform, triangular, normal, and lognormal distributions. It also covers discrete probability distributions and their pdfs, as well as the normal distribution and its properties.

Typology: Study notes

Pre 2010

1 / 10

Download Understanding Probability Distributions: Continuous, Discrete Variables & Common and more Study notes Environmental Science in PDF only on Docsity! 1 Module 1: Review of Basic Statistical Concepts 1.1 Understanding Probability Distributions, Parameters and Statistics 4/18/2002 Module 2 Distributions of Data A variable that can take on any value in a range is called a continuous variable. • Example: The concentration of a contaminant in water samples A variable that can take on only certain values is called discrete. • Example: The number of animals visiting a contaminated site in a single day 2 4/18/2002 Module 3 Distributions of Data A probability distribution describes the values that a variable can take on and the probabilities associated with those values. We use probability density functions (pdfs) to describe these distributions. We can also use cumulative density functions (cdfs) For continuous variables, common distributions are the uniform, the triangular, the normal, and the lognormal 4/18/2002 Module 4 Examples of Continuous Probability Density Functions (pdfs) Uniform Distribution 0 0.01 0.02 0.03 0.04 0.05 0.06 0 10 20 30 40 50 60 70 80 90 Value PR OB AB IL IT Y Tr iangular Distribution 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1 10 20 30 40 50 60 70 80 90 Value P R O B A B IL IT Y Normal Distribution 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 -2 .6 -0 .9 0.8 2 .5 4.2 5 .9 7.6 9.3 11 .0 12 .7 P R O B A B IL IT Y Lognormal Distribution 0 0.05 0.1 0.15 0.2 0.25 0.3 0.3 1.3 2 .4 3.5 4 .5 5.6 6 .7 7.7 8 .8 9.9 P R O B A B IL IT Y 5 4/18/2002 Module 9 Examples of Discrete Probability Distribution Functions (pdfs) Toss of a Fair Coin 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 Result Pr ob ab ili ty TailHead Roll of a Fair Die 0 0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 6 General Example of a Discrete pdf 0 0.1 0.2 0.3 1 2 3 4 5 6 7 8 9 Value P ro b ab ili ty 4/18/2002 Module 10 The Normal Distribution The normal distribution is the bell-shaped curve. Many things that occur in nature follow a normal distribution. Some characteristics: • It has two parameters: the mean mu (µ) and the standard deviation sigma (σ) 6 4/18/2002 Module 11 The Normal Distribution Some characteristics: • The standard normal has µ=0 and σ=1 • Any normal distribution can be transformed into a standard normal by Z=(X- µ)/σ • About 68% of the probability of a normal lies within plus and minus one standard deviation from the mean • About 95% lies within plus and minus 2 standard deviations from the mean • 99.7% lies within plus and minus 3 standard deviations from the mean 4/18/2002 Module 12 Using the Table of the Normal Distribution Tables such as Table B1 in Manly relate values of Z to the probability (area) under the normal pdf from zero to that value. • Example: The probability of a value sampled randomly from the standard normal distribution falling between the mean and one standard deviation above it is found by looking up the probability in the table associated with 1.00. It is 0.341. So, the probability of a value falling within 1 standard deviation from the mean is double that or 0.682. Likewise, the probability of a value falling within plus and minus 2 standard deviations is 2* 0.477=0.954. 7 4/18/2002 Module 13 The Binomial Distribution Applies in a situation where there are two possible outcomes (success and failure) and the probability of success is constant. Example: Failure is defined to be contamination above a regulatory limit. Assume contamination is uniformed dispersed throughout an area such as a lake and n samples are collected. There will be variability in the amount of measured contamination in the samples due to sampling and measurement errors. There is a probability p that each of the samples will show contamination above the limit. 4/18/2002 Module 14 The Student’s t Distribution The Student's t distribution is similar to the normal but with fatter tails. It is used when the true population standard deviation is not known (most of the time). The exact shape of the t distribution is controlled by the number of data points used to calculate the sample standard deviation. When n is small, the distribution is wide. When n gets large, the estimate of σ is good and the t distribution approaches the shape of a normal distribution. The term for this index is called degrees of freedom (df). For use with the t distribution, df = n-1.