Probability Distributions (Part 1) PDF

 Probability Distributions (Part 1) Week 2: AMAT 131 Statistical Methods and Experimental Design EODiamante DMPCS, CSM, UP Mindanao Recall: Probability Distributions The...

 Probability Distributions (Part 1) Week 2: AMAT 131 Statistical Methods and Experimental Design EODiamante DMPCS, CSM, UP Mindanao Recall: Probability Distributions The probability structure of a random variable, 𝑦, is described by its probability distribution. ▪ If 𝑦 is discrete, 𝑝(𝑦) is called the probability mass function of y ▪ If 𝑦 is continuous, 𝑝(𝑦) is called the probability density function of y Probability Distribution ▪ Probability Distribution describes the probability structure of a random variable ▪ It consists of the possible values of the random variable with its corresponding probabilities. ▪ Can be represented by a table, graph, or formula. Discrete Probability Distribution ▪ Values of a discrete random variable and their corresponding probabilities. ▪ Probabilities are determined theoretically or by observation. ▪ Probability mass function, denoted by 𝑝 𝑦 = 𝑃(𝑌 = 𝑦) for all 𝑦 Properties of Discrete Probability Distribution 1. 0 ≤ 𝑝 𝑦 ≤ 1 for all values of 𝑦. 2. 𝑃 𝑌 = 𝑦 = 𝑝(𝑦) for all values of 𝑦. 3. σ𝑦 𝑝(𝑦) = 1 Continuous Probability Distribution ▪ Values of a continuous random variable and their corresponding probabilities. ▪ Probability density function, denoted by 𝑓(𝑦) ▪ PDF is a theoretical model for the frequency distribution of a population measurements. Properties of Continuous Probability Distribution 1. 𝑓 𝑦 ≥ 0 for all 𝑦 𝑏 2. 𝑃 𝑎 ≤ 𝑌 ≤ 𝑏 = ‫𝑓 𝑎׬‬ 𝑦 𝑑𝑦 +∞ 3. ‫׬‬−∞ 𝑓 𝑦 𝑑𝑦 = 1 Mean of a Probability Distribution The mean, 𝜇, of a probability distribution is a measure of its central tendency/location. We define it as: 𝜇 = ෍ 𝑦 ∙ 𝑝(𝑦) 𝑓𝑜𝑟 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑎𝑙𝑙 𝑦 +∞ 𝜇=න 𝑦 ∙ 𝑓 𝑦 𝑑𝑦 𝑓𝑜𝑟 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 −∞ The mean is also known as the expected value 𝐸(𝑌) Variance of a Probability Distribution 2 The variance, 𝜎 , of a probability distribution is a measure of its variability/dispersion. We define it as: 𝜎2 = ෍ 𝑦 − 𝜇 2 ∙ 𝑝(𝑦) 𝑓𝑜𝑟 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑎𝑙𝑙 𝑦 +∞ 2 𝜎=න 𝑦−𝜇 ∙ 𝑓 𝑦 𝑑𝑦 𝑓𝑜𝑟 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 −∞  Variance of a Probability Distribution The variance can be expressed entirely in terms of expectation, because 2 2 2 2 𝜎 = 𝑉(𝑌) = 𝐸 𝑌 − 𝜇 =𝐸 𝑌 − 𝐸 𝑌 The standard deviation of 𝑌 is the positive square root of 𝑉(𝑌) Example 1: Construct a probability distribution and compute the expected value, variance, and standard deviation of a random experiment: number of heads in tossing three (3) coins.  Binomial Experiment 1. The experiment has a fixed number, 𝑛, of identical trials. 2. Each trial has two (2) possible outcomes: success or failure. 3. Trials are independent from one another. 4. Probability of success 𝑝 on a single trial remains the same. Probability of failure is 1 − 𝑝. 5. Random variable 𝑌 is the number of successes observed during the 𝑛 trials.  Binomial Probability Distribution A random variable 𝑌 is said to have a binomial distribution on 𝑛 trials with success probability 𝑝 if and only if: 𝑛 𝑦 𝑛−𝑦 𝑃 𝑌 =𝑝 𝑦 = 𝑦 𝑝 𝑞 P(S) – prob. of success P(F) – prob. of failure 𝑛 – no. of trials 𝑦- no. of successes in 𝑛 trials 𝑝(𝑦) – binomial probability mass function Example 2: A coin is tossed three times. Find the probability of getting exactly two heads. Is this experiment binomial? Example 3: McJollibee plans to open 6 branch outlets in Davao City. From experience, they know that 20% of the new outlets will experience difficulty in penetrating the sales region and fail. Using this estimate, determine the probability distribution for the number of these outlets which will succeed. Example 3: Example 3: What is the probability that ▪ at least 4 of the outlets will succeed? ▪ at most 2 will succeed?  Binomial Probability Distribution The mean, variance, and standard deviation is: 𝜇 = 𝑛𝑝 𝜎 2 = 𝑛𝑝𝑞 𝜎 = 𝑛𝑝𝑞  Poisson Probability Distribution 1. No. of trials 𝑛 is large and the success 𝑝 is small (occur over a period of time). 2. Can be used when a density of items is distributed over a given time, area, volume, etc. 3. Random variable 𝑌 is the number of occurrences of an event in an interval of time. 4. Occurrence of events is independent. The total number of occurrences has a binomial distribution.  Poisson Probability Distribution The probability of 𝑌 occurrences in an interval of time for a variable when 𝜆 is the mean number of occurrences per unit is: 𝑒 −𝜆 𝜆𝑦 𝑝 𝑦 = 𝑦! y– 0,1,2,… 𝜆 – average value of 𝑌 𝑒- Euler constant  Poisson Probability Distribution The mean, variance, and standard deviation is: 𝜇=𝜆 𝜎2 = 𝜆 𝜎= 𝜆 Example 4: If there are 200 typographical errors randomly distributed in a 500-page manuscript, find the probability that a given page contains exactly three errors. Example 4: Example 5: A sales firm receives, on average, 3 calls per hour on its toll-free number. For any given hour, find the probability that it will receive: ▪ At most 3 calls. ▪ At least three calls ▪ 5 or more calls Example 5: At most 3 calls. Example 5: ▪ At least three calls Example 5: ▪ 5 or more calls  Questions?  Next meeting: Probability Distributions (Part 2) Multinomial, Normal, and other distributions