# Lecture_09 - 1 ISE 690 Statistical Methods for Engineers...

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1 Industrial and Systems and Engineering Management Department Fall 15 Dilcu Helvaci Barnes (Email: ISE 690 Statistical Methods for Engineers Joint Probability Distributions

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2 Joint Probability Distributions In Chapters 3 and 4, we developed probability models for a single random variable. Many problems in probability and statistics involve several random variables simultaneously in which more than one random variable will be of interest to an investigator. The measurements might not be independent, so the joint relationship between these measurements becomes important.
3 Joint Probability Mass Function Consider two production lines that manufacture a certain item. The production rates for both lines vary randomly from day to day. Line 1 has a capacity of 4 units per day while line II has a capacity of 3 units per day. Further, both lines produce at least one unit on any given day. Let X 1 = No. of units produced by line I, and X 2 = No. of units produced by line II per day. The joint probability (Pr) distribution (JPD) of the bivariate vector is given below

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4 Joint Probability Mass Function =Number of units produced by = Number of units produced by the Line II 1 2 3 f 1 0.01 0.05 0.04 2 0.05 0.10 0.10 3 0.10 0.15 0.10 4 0.04 0.15 0.11 1 2 3 1 0.01 0.05 0.04 2 0.05 0.10 0.10 3 0.10 0.15 0.10 4 0.04 0.15 0.11
5 Joint Probability Mass Function The joint probability mass function of the discrete random variables X and Y, denoted as satisfies the following properties: Just as the probability mass function of a single random variable X is assumed to be zero at all values outside the range of X, so the joint probability mass function of X and Y is assumed to be zero at values for which a probability is not specified.

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6 Joint Probability Density Function The joint probability distribution of two continuous random variables X and Y can be specified by providing a method for calculating the probability
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• Fall '15
• DilcuBarnes
• Probability, Probability theory, probability density function, Systems and Engineering Management, Helvaci Barnes

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