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(5) 3. A die with three sides (1, 2, 3 is tossed two times. Let X...
Problem 2. (15 pts) A fair die is tossed 20 times in succession. Let Y be...
Problem 2. (15 pts) A fair die is tossed 20 times in succession. Let Y be the total number of sixes that occur, and let X be the number of sixes occurring in the first 5 tosses. Determine the conditional probability mass function P(X r|Y ).
5. (15 pts) a) A coin is tossed 5 times. Let X be the number of...
5. (15 pts) a) A coin is tossed 5 times. Let X be the number of Heads on the first 4 tosses and Y be the number of Heads on the last three tossed. Find the joint probabilities Pij = P(X = 1, Y = j) for all relevant i and j. Find the marginal probabilities pit and p+, for all relevant i and j. b) Find the value of A that would make the function Af(x,y) a PDF. Where...
1. A fair coin is tossed three times. Let A be the event that there are...
1. A fair coin is tossed three times. Let A be the event that there are at least two heads in the three tosses and let B be the event that there are exactly two heads among the three tosses. a. Draw the complete tree diagram for this experiment. [3] b. What are the sample space and probability function for this experiment? [5] c. Compute P(A), P(B), P(A|B), and P(B|A). [7]
1. A fair coin is tossed three times. Let A be the event that there are...
1. A fair coin is tossed three times. Let A be the event that there are at least two heads in the three tosses and let B be the event that there are exactly two heads among the three tosses. a. Draw the complete tree diagram for this experiment. [3] b. What are the sample space and probability function for this experiment? [5] c. Compute P(A), P(B), P(A|B), and P(B|A). [7]
Extra: Let X, Y, Z be results of three independent tosses of a fair die. (a)...
Extra: Let X, Y, Z be results of three independent tosses of a fair die. (a) Find the covariance of the random variables W=2X-3Y + Z (b) Find the correlation coefficient of W and V. and V=X-2Y-Z
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's...
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's and let y be the number of 2's obtained. What is the joint PMF of X and Y? (b) Let X1, X2, X3 be independent random variables, uniformly distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is the middle of the three values). Find the conditional CDF of X1, given that Y = 0.5. Under this conditional distribution, is...
1. A fair coin is tossed three times. Let A be the event that there are...
1. A fair coin is tossed three times. Let A be the event that there are at least two heads in the three tosses and let B be the event that there are exactly two heads among the three tosses. a. Draw the complete tree diagram for this experiment. [3] b. What are the sample space and probability function for this experiment? (5) c. Compute P(A), P(B), P(A|B), and P(BA). [7] 2. Let U be a continuous random variable with...
Two fair 6-sided dice are tossed. Let X denote the number appearing on the first die...
Two fair 6-sided dice are tossed. Let X denote the number appearing on the first die and let y denote the number appearing on the second die. Show that X, Y are independent by showing that P(X = x, Y = y) = P(X = x) x P(Y = y) for all (x,y) pairs.
2. (a) Die #1 has 6 sides numbered 1, . . . , 6 and die...
2. (a) Die #1 has 6 sides numbered 1, . . . , 6 and die #2 has 8 sides numbered 1, . . . , 8. One of these two dice is chosen at random and rolled 10 times. Find the conditional probability that you have selected die #1 given that precisely three 1’s were rolled. (b) Let X and Y be independent Poisson random variables with mean 1. Are X − Y and X + Y independent? Justify...
A fair tetrahedron (four-sided die) is rolled twice. Let X be the random variable denoting the...
A fair tetrahedron (four-sided die) is rolled twice. Let X be the random variable denoting the total number of dots in the outcomes, and Y be the random variable denoting the maximum in the two outcomes. Thus if the outcome is a (2, 3) then X = 5 while Y = 3. (a) What are the ranges of X and Y ? (b) Find the probability mass function (PMF) of X and present it graphically. Describe the shape of this...
Problem 2. (15 pts) A fair die is tossed 20 times in succession. Let Y be the total number of sixes that occur, and let X be the number of sixes occurring in the first 5 tosses. Determine the conditional probability mass function P(X r|Y ).
5. (15 pts) a) A coin is tossed 5 times. Let X be the number of Heads on the first 4 tosses and Y be the number of Heads on the last three tossed. Find the joint probabilities Pij = P(X = 1, Y = j) for all relevant i and j. Find the marginal probabilities pit and p+, for all relevant i and j. b) Find the value of A that would make the function Af(x,y) a PDF. Where...
1. A fair coin is tossed three times. Let A be the event that there are at least two heads in the three tosses and let B be the event that there are exactly two heads among the three tosses. a. Draw the complete tree diagram for this experiment. [3] b. What are the sample space and probability function for this experiment? [5] c. Compute P(A), P(B), P(A|B), and P(B|A). [7]
1. A fair coin is tossed three times. Let A be the event that there are at least two heads in the three tosses and let B be the event that there are exactly two heads among the three tosses. a. Draw the complete tree diagram for this experiment. [3] b. What are the sample space and probability function for this experiment? [5] c. Compute P(A), P(B), P(A|B), and P(B|A). [7]
Extra: Let X, Y, Z be results of three independent tosses of a fair die. (a) Find the covariance of the random variables W=2X-3Y + Z (b) Find the correlation coefficient of W and V. and V=X-2Y-Z
(a) Consider four independent rolls of a 6-sided die. Let X be the number of l's and let y be the number of 2's obtained. What is the joint PMF of X and Y? (b) Let X1, X2, X3 be independent random variables, uniformly distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is the middle of the three values). Find the conditional CDF of X1, given that Y = 0.5. Under this conditional distribution, is...
1. A fair coin is tossed three times. Let A be the event that there are at least two heads in the three tosses and let B be the event that there are exactly two heads among the three tosses. a. Draw the complete tree diagram for this experiment. [3] b. What are the sample space and probability function for this experiment? (5) c. Compute P(A), P(B), P(A|B), and P(BA). [7] 2. Let U be a continuous random variable with...
Two fair 6-sided dice are tossed. Let X denote the number appearing on the first die and let y denote the number appearing on the second die. Show that X, Y are independent by showing that P(X = x, Y = y) = P(X = x) x P(Y = y) for all (x,y) pairs.
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