MATH30-6-Lecture-5

# MATH30-6-Lecture-5 - Joint Probability Distributions...

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MATH30-6 Probability and Statistics Joint Probability Distributions

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Objectives At the end of the lesson, the students are expected to Use joint probability mass functions and joint probability density functions to calculate probabilities; Calculate marginal and conditional probability distributions from joint probability distributions; and Interpret and calculate covariances and correlations between random variables.
Joint Probability Mass Function The joint probability mass function of the discrete random variables X and Y , denoted as f XY ( x , y ), satisfies (1) f XY ( x , y ) ≥ 0 (2) (3) f XY ( x , y ) = P ( X = x , Y = y ) (5-1) Sometimes referred to as the bivariate probability distribution or bivariate distribution of the random variables P ( X = x and Y = y ) is usually written as P ( X = x , Y = y ).

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Joint Probability Mass Function Examples: 5-1/156 Mobile Response Time The response time is the speed of page downloads and it is critical for a mobile Web site. As the response time increases, customers become more frustrated and potentially abandon the site for a competitive one. Let X denote the number of bars of service, and let Y denote the response time (to the nearest second) for a particular user and site.
Joint Probability Mass Function By specifying the probability of each of the points in Fig. 5-1, we specify the joint probability distribution of X and Y . Similarly to an individual random variable, we define the range of the random variables ( X , Y ) to be the set of points ( x , y ) in two-dimensional space for which the probability that X = x and Y = y is positive.

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Joint Probability Mass Function 3.14/95 Two refills for a ballpoint pen are selected at random from a box that contains 3 blue refills, 2 red refills, and 3 green refills. If X is the number of blue refills and Y is the number of red refills selected, find (a) the joint probability function f ( x , y ), and (b) P [( X , Y ) A ], where A is the region {( x , y )| x + y ≤ 1}.
Joint Probability Density Function A joint probability density function for the continuous random variables X and Y , denoted as f XY ( x , y ), satisfies the following properties: (1) f XY ( x , y ) ≥ 0 for all x , y (2) (3) For any region R of two-dimensional space, (5-2)

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Joint Probability Density Function Examples: 5-2/158 Server Access Time Let the random variable X denote the time until a computer server connects to your machine (in milliseconds), and let Y denote the time until the server authorizes you as a valid user (in milliseconds). Each of these random variables measures the wait from a common starting time and X < Y . Assume that the joint probability density function for X and Y is f XY ( x , y ) = 6 × 10 −6 e −0.001 x − 0.002 y for x < y
Joint Probability Density Function The region with nonzero probability is shaded in Fig. 5-4. The property that this joint probability density function integrates to 1 can be verified by the integral of f XY ( x , y )over this region as follows:

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Joint Probability Density Function
Joint Probability Density Function

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Joint Probability Density Function The probability that X < 1000 and Y < 2000 is determined as the integral over the darkly shaded region in Fig 5-5.
Joint Probability Density Function P ( X ≤ 1000, Y ≤ 2000) = 0.003(316.738 − 11.578) = 0.915

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