1. In a restaurant there are 20 baseball players: 5 Mets players and 15 Yankees players. Of these, 4 Mets players and 4 Yankees players are drinking beer.
(a) A Yankees player is randomly selected. What is the probability that he is drinking beer?
(b) Are these two events (being a Yankees player and drinking beer) statistically independent? Why or why not - be specific about what information you used to answer this question?
2.You are given the following probability distribution of X, GPA of student, and Y, hours of studying each week by student:
|
|
|
X (GPA)
|
|
|
|
2
|
3
|
4
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Y (hours
of studying
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20
|
0.2
|
0.1
|
0.0
|
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30
|
0.1
|
0.2
|
0.1
|
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40
|
0.0
|
???
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0.2
|
(a) Find the P(X=4,Y<30), and write in words what this probability means.
(b) Find the marginal distributions of X and Y.
(c) Find P(Y<30), and write in words what this probability means.
(d) Find the expected value of X.
(e) Find the variance of X.
(f) What is the conditional distribution of Y given that X=3?
(g) Find E(Y|X = 3) and write in words what this expectation means.
(h) Are X and Y independent? Why or why not - be specific about what information you used to answer this question.
3. You are told that X and Y are random variables, that they are INDEPENDENT (and hence uncorrelated with each other), and that
X ~ N (2, 1) and Y ~ N (4, 8).
In a.-d. below, you are given expressions or formulas for W where W is a mathematical function of X and/or Y. For each of these, you need to find the distribution for the random variable W. This requires finding the Expected Value, E(W), and the Variance, var(W), and writing your answer using the notation W ~ N (E(W), var(W)).
(a) W = (1 / 5) X
(b) W = X + Y
(c) W = X - Y
(d) W = 2X - 4Y
4. Repeat part (d) ONLY of Question 3, but now assume that X and Y are positively correlated with a correlation coefficient of +1.0.Find the distribution for the random variable W.