Two football teams will participate in an international tournament. The estimated probabilities for the numbers of...
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Two football teams will participate in an international tournament. The estimated probabilities for the numbers of goals these teams will score are shown in the following table: 0 number of 0 0.098 goals scored 1 0.142 by team 1 2 0.094 3 0.067 4 5 number of goals scored by team 2 4 1 0.043 0.130 2 0.035 0.028 3 0.012 0.007 0.008 0.008 0.008 0.004 0.004 0.000 0.000 0.118 0.063 0.043 0.028 0.008 0.016 0.008 0.004 0.000 0.016 0.000 0.003 0.000 5 0.003 0.002 0.000 0.000 0.000 0.000 Denote with X; the random variable that counts the number of goals that team i E {1,2} will score. a. Calculate P[X 3, X > 2]. b. Calculate Fx,x (2, 1). c. Find the marginal probability mass function of both X and X. d. Are X and X stochastically independent? Why (not)? e. What is the probability mass function of X if it is known that team 2 will score two goals? j. Calculate the variances of both X and X. k. Calculate the covariance between X and X2. 1. Calculate the correlation between X and X. m. Calculate E[XX = 3]. n. Calculate E[E[3X1X]]. f. Find the probability mass function of X conditioned on the event {X = x} for every x {0, 1, 2, 3, 4, 5). g. Apply results of part f in the law of total probability to calculate P[X = 0] and P[X = 3]. h. Use results of part f in Bayes' formula to obtain P[X = 21X = 3]. i. Calculate the expectations of both X and X. Two football teams will participate in an international tournament. The estimated probabilities for the numbers of goals these teams will score are shown in the following table: 0 number of 0 0.098 goals scored 1 0.142 by team 1 2 0.094 3 0.067 4 5 number of goals scored by team 2 4 1 0.043 0.130 2 0.035 0.028 3 0.012 0.007 0.008 0.008 0.008 0.004 0.004 0.000 0.000 0.118 0.063 0.043 0.028 0.008 0.016 0.008 0.004 0.000 0.016 0.000 0.003 0.000 5 0.003 0.002 0.000 0.000 0.000 0.000 Denote with X; the random variable that counts the number of goals that team i E {1,2} will score. a. Calculate P[X 3, X > 2]. b. Calculate Fx,x (2, 1). c. Find the marginal probability mass function of both X and X. d. Are X and X stochastically independent? Why (not)? e. What is the probability mass function of X if it is known that team 2 will score two goals? j. Calculate the variances of both X and X. k. Calculate the covariance between X and X2. 1. Calculate the correlation between X and X. m. Calculate E[XX = 3]. n. Calculate E[E[3X1X]]. f. Find the probability mass function of X conditioned on the event {X = x} for every x {0, 1, 2, 3, 4, 5). g. Apply results of part f in the law of total probability to calculate P[X = 0] and P[X = 3]. h. Use results of part f in Bayes' formula to obtain P[X = 21X = 3]. i. Calculate the expectations of both X and X.
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The Legal Environment of Business A Critical Thinking Approach
ISBN: 978-0132664844
6th Edition
Authors: Nancy K Kubasek, Bartley A Brennan, M Neil Browne
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