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Problem 5 Two fair dice are tossed, and ((X,Y) denote the number of spots on the...
Two fair dice are tossed independently, and the pair, (X, Y), denote the number of spots...
Two fair dice are tossed independently, and the pair, (X, Y), denote the number of spots on the first and on the second dice. Consider two random variables: U = X + Y and W = X-Y! 1. Find the distribution of U. 2. Find the distribution of W. 3. Find the conditional distribution of W, given that U 6
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.
A fair coin is tossed. If the toss results in a head, then one die is...
A fair coin is tossed. If the toss results in a head, then one die is thrown, while if the toss results in a tail, then two dice are thrown. Let X denote the random variable that counts the number of spots showing on the thrown die or dice. The values that X can assume are the positive integers from 1 to 12 inclusive. Find the following probabilities. Your answers should be whole numbers or fractions in lowest terms. Pr(X=1)...
2.1 Let Y denote the number of "heads” that occur when two coins are tossed. a....
2.1 Let Y denote the number of "heads” that occur when two coins are tossed. a. Derive the probability distribution of Y. b. Derive the cumulative probability distribution of Y. c. Derive the mean and variance of Y.
Problem 5. Suppose two dice are tossed and the numbers on the upper faces are observed....
Problem 5. Suppose two dice are tossed and the numbers on the upper faces are observed. Let S denote the set of all possible pairs that can be observed. The pairs can be listed, for example, by letting (2, 3) denote that a 2 was observed on the first die and a 3 on the second. Define the following subsets of S .A The number on the second die is odd. · B: The sum of the two numbers is...
3. (a) A fair dice is tossed 6 times. Suppose A is the event that the...
3. (a) A fair dice is tossed 6 times. Suppose A is the event that the number of occurrences of an even digit equals the number of occurrences of an odd digit, while B is the event that at most three odd digits will occur i. Determine with reason if the events A and B are mutually exclusive. ii. Determine the probabilities of the events A and B. Are the events A and B independent? b) Suppose a fair coin...
In a particular game, a fair die is tossed. If the number of spots showing is...
In a particular game, a fair die is tossed. If the number of spots showing is a six, you win $6, if the number of spots showing is a five, you win $3, if the number of spots showing is 4, you win $2, and if the number of spots showing is 3, you win $1. If the number of spots showing is 1 or 2, you win nothing. You are going to play the game twice. Let X be...
Roll two fair six-sided dice, and let X, Y denote the first and the second numbers....
Roll two fair six-sided dice, and let X, Y denote the first and the second numbers.If Z=max {X, Y}, find- E(Z)- V(Z)If Z=|X-Y|, find- E(Z)- V(Z)
Consider a roll of a pair of fair dice. Let X = absolute value of the...
Consider a roll of a pair of fair dice. Let X = absolute value of the difference of the two dice. What are the possible values that X can take on? Derive both the mass function and the distribution function for X.
2. Assume two fair dice are rolled. Let X be the number showing on the first die and number showing on the second die. (a) Construct the matrix showing the joint probability mass function of the pair...
2. Assume two fair dice are rolled. Let X be the number showing on the first die and number showing on the second die. (a) Construct the matrix showing the joint probability mass function of the pair X,Y. (b) The pairs inside the matrix corresponding to a fixed value of X - Y form a straight line of entries inside the matrix. Draw those lines and use them to construct the probability mass function of the random variable X-Y- make...
Two fair dice are tossed independently, and the pair, (X, Y), denote the number of spots on the first and on the second dice. Consider two random variables: U = X + Y and W = X-Y! 1. Find the distribution of U. 2. Find the distribution of W. 3. Find the conditional distribution of W, given that U 6
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.
2.1 Let Y denote the number of "heads” that occur when two coins are tossed. a. Derive the probability distribution of Y. b. Derive the cumulative probability distribution of Y. c. Derive the mean and variance of Y.
Problem 5. Suppose two dice are tossed and the numbers on the upper faces are observed. Let S denote the set of all possible pairs that can be observed. The pairs can be listed, for example, by letting (2, 3) denote that a 2 was observed on the first die and a 3 on the second. Define the following subsets of S .A The number on the second die is odd. · B: The sum of the two numbers is...
3. (a) A fair dice is tossed 6 times. Suppose A is the event that the number of occurrences of an even digit equals the number of occurrences of an odd digit, while B is the event that at most three odd digits will occur i. Determine with reason if the events A and B are mutually exclusive. ii. Determine the probabilities of the events A and B. Are the events A and B independent? b) Suppose a fair coin...
In a particular game, a fair die is tossed. If the number of spots showing is a six, you win $6, if the number of spots showing is a five, you win $3, if the number of spots showing is 4, you win $2, and if the number of spots showing is 3, you win $1. If the number of spots showing is 1 or 2, you win nothing. You are going to play the game twice. Let X be...
Roll two fair six-sided dice, and let X, Y denote the first and the second numbers.If Z=max {X, Y}, find- E(Z)- V(Z)If Z=|X-Y|, find- E(Z)- V(Z)
2. Assume two fair dice are rolled. Let X be the number showing on the first die and number showing on the second die. (a) Construct the matrix showing the joint probability mass function of the pair X,Y. (b) The pairs inside the matrix corresponding to a fixed value of X - Y form a straight line of entries inside the matrix. Draw those lines and use them to construct the probability mass function of the random variable X-Y- make...
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