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381 Introduction to Probability Distributions QSCI 381 – Lecture 12 (Larson and Farber, Sect 4.1) 381 Learning objectives Become comfortable with variable definitions Create and use probability distributions Random Variables-I 381 A X represents a numerical value associated with each outcome of a probability experiment. “Random” implies that there is an element of chance involved. Typical random variables: Time that a tagged animal is at large. Number of offspring produced by a female Panda bear in one year / over her lifespan. Length / weight of a sampled fish Random Variables - II 381 Notation: We will use upper case letters to denote random variables and lower case letters to denote particular values of random variables. The probability that the random variable X has value x is therefore written as P (X=x). Random Variables - III 381 A random variable is: if it has a countable number of possible outcomes that can be listed. if it has an uncountable number of possible outcomes (or the number of outcomes cannot be listed). Continuous random variables can (at least conceptually) be measured to any degree of accuracy. It is important to be able to distinguish between continuous and discrete random variables as we will analyze them differently Random Variables - III 381 The number of scales forming the lateral line of a fish? Max swimming speed of a blue whale? Discrete random variable Continuous random variable Reproductive events of a Fathead minnow each year? Discrete random variable 381 Discrete and Continuous Random Variables Which of the following random variables that one might encounter on a fishery survey are discrete and which are continuous? Number of animals caught in a haul. Actual length of the 5th animal in the haul. Length of the 5th animal in the haul as measured by an observer. Duration of the 7th haul. Number of hauls before a shark is caught. The weight of the haul. The ratio of marketable species to unmarketable species. Probability Distributions (Discrete-I) 381 A lists each possible value the random variable can take, together with its probability. From our earlier discussion of probability, it must be true that: The probability of each value of the discrete random variable must lie between 0 and 1. The sum of all the probabilities must equal 1. Probability Distributions (Discrete-II) 381 Mathematically: Let the possible values for the random variable be: {xi : i 1,..., N } The probability of By the first condition: each value of the 0 P( xi ) 1 By the second condition: Upper summation limit N P( x ) 1 Lower summation limit i 1 i discrete random variable is between 0 and 1, inclusive. The sum of all the probabilities is 1. Probability Distributions (Discrete-III) 381 0 Consider the random variable “what is the maturity state of the next bowhead whale we observe during a survey?” This random variable is discrete because it has three outcomes: 1-calf; 2-immature; 3mature. 0.5 1 Probability Distributions (Discrete-III) 381 0 Consider the random variable “what is the maturity state of the next bowhead whale we observe during a survey?” This random variable is discrete because it has three outcomes: 1-calf; 2-immature; 3mature. 0.5 Does this satisfy both discrete probability conditions? 1 Probability Distributions (Discrete-III) 381 Consider the random variable “what is the maturity state of the next bowhead whale we observe during a survey?” This random variable is discrete because it has three outcomes: 1-calf; 2-immature; 3mature. The probabilities for each outcome are: P(calf) = 0.05; P(immature)=0.52; P(mature)=0.43. The probabilities satisfy the requirements for a discrete probability distribution. Probability Distributions (Discrete-IV) 381 Discrete probability distributions are often represented using histograms. 0.6 0.52 0.5 0.43 Probability 0.4 0.3 0.2 0.1 0.05 0 Calf Immature Maturity State Mature Example-I 381 A fish is tagged and released. Given that it is recaptured within 10 days of release, the following are the probabilities of recapture by day: 1 2 3 4 5 0.02 0.05 0.1 0.2 0.3 6 7 8 9 10 0.23 0.05 0.02 0.02 0.01 Verify that this is a discrete random variable. Means, Variances and Standard Deviations-I 381 The given by: of a discrete random variable is N xi P ( xi ) i 1 Each value of x is multiplied by its corresponding probability and the products are added. Find the mean number of days before the fish in Example I is recaptured. Note that the mean need not be an integer. Excel Hint: Use the “Sumproduct(A1:J1,A2:J2)” command to find the mean. Means, Variances and Standard Deviations-II 381 The of a discrete random variable is given by: N ( xi ) 2 P( xi ) 2 i 1 and the by: 2 Calculation of Variance (Example) 381 x P( x) x ( x )2 1 2 3 0.16 0.22 0.28 -1.94 -0.94 0.06 3.764 0.884 0.004 4 5 0.20 0.14 1.06 2.06 1.124 4.244 Hint: First find the mean, =2.94 P( x)( x )2 0.602 0.194 0.001 0.225 0.594 1.616 The variance is the summation of Expected Value 381 The of a discrete random variable is equal to the mean of the random variable, i.e.: N E ( X ) X xi P( xi ) i 1