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Statitics%20lecture%209 - Lecture Notes 1 MSc in...

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Lecture Notes 1 MSc in Operational Research Introduction to Statistics Lecture 9 Bivariate Probability Distributions Introduction The intersection of two or more events is frequently of interest to an experimenter. For example, the gambler playing blackjack is interested in the event of drawing both an ace and a face card from a fifty-two-card deck. The biologist, observing the number of animals surviving in a litter, is concerned with the intersection of these events: A: The litter contains n animals B: y animals survive. Similarly, the observation of both height and weight on an individual represents the intersection of a specific pair of height-weight measurements. Most important to statisticians are the intersections that occur when sampling. Suppose that Y 1 ,Y 2 , ..., Y n denote the outcomes on n successive trials of an experiment. For example, this sequence could represent the weights of n people or the measurements of n physical characteristics of a single person. A specific set of outcomes, or sample measurements, may be expressed in terms of the intersection of the n events ), y Y ( 1 1 = ), y Y ( 2 2 = …. ), y Y ( n n = which we will denote as ). y ...... , y , y , y ( n 3 2 1 Then to make inferences about the population from which the sample was drawn, we will wish to calculate the probability of the intersection ). y ...... , y , y , y ( n 3 2 1 A review of the role probability plays in making inferences, emphasizes the need for acquiring the probability of the observed sample or, equivalently, the probability of the intersection of a set of numerical events. Knowledge of this probability is fundamental to making an inference about the population from which the sample was drawn. Indeed, this need motivates the discussion of multivariate probability distributions. Bivariate Probability Distributions Many random variables can be defined over the same sample space. For example, consider the experiment of tossing a pair of dice. The sample space contains thirty-six sample points, corresponding to the mn = (6)(6) = 36 ways which numbers may appear on the faces of the dice. Any one of the following random variables could be defined over the sample space and might be of interest to the experimenter: Y,: The number of dots appearing on die 1. Y 2 : The number of dots appearing on die 2. Y 3 : The sum of the number of dots on the dice.
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Lecture Notes 2 Y 4 : The product of the number of dots appearing on the dice. The thirty-six sample points associated with the experiment are equiprobable and corresponding to the thirty-six numerical events ). y , y ( 2 1 Thus throwing a pair of is would be the simple event (1, 1). Throwing a 2 on die 1 and a 3 on die 2 would be the simple event (2, 3). Because all pairs ) y , y ( 2 1 occur with the same relative frequency, we would assign a probability of 1/36 to each sample point. For this simple example the intersection ) y , y ( 2 1 contains only one sample point. Hence the bivariate probability function is 36 1 ) y , y ( p 2 1 = y l = 1, 2,..., 6; y 2 = 1, 2,..., 6 Definition Let Y 1 and Y 2 be discrete random variables. The joint (or bivariate) probability distribution
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