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 your answer.
(c) Let X be a geometric random variable with parameter p. Find E[min(X, 5)].
Homework Answers
You have a pair of 4-sided dice. The four sides of each die are numbered 1,...
You have a pair of 4-sided dice. The four sides of each die are numbered 1, 2. 3, and 4. Each time the pair of dice is rolled, you add the numbers from each die. Out of all the possible ways the dice can land, how many of them give you a sum of 5? Number How many ways give you a sum of 8? Number What is the probability of rolling a sum of 7 with these dice? Number
A single six-sided die, whose faces are numbered 1 to 6, is rolled n times. The...
A single six-sided die, whose faces are numbered 1 to 6, is rolled n times. The die is fair, each face is equally likely to land upward when the die is rolled. Let X be the number of times that the number on the upward face of the die is 1. Find the mean and the standard deviation of the random variable X.
Two identical fair 6-sided dice are rolled simultaneously. Each die that shows a number less than...
Two identical fair 6-sided dice are rolled simultaneously. Each die that shows a number less than or equal to 4 is rolled once again. Let X be the number of dice that show a number less than or equal to 4 on the first roll, and let Y be the total number of dice that show a number greater than 4 at the end. (a) Find the joint PMF of X and Y . (Show your final answer in a...
(1 point) Suppose that a single die with 7 sides (numbered 1, 2, 3, ... ,...
(1 point) Suppose that a single die with 7 sides (numbered 1, 2, 3, ... , 7) is rolled twice. What is the probability that the sum of the two rolls equals 3 ?
Problem 2: A player chooses one of the numbers 1 through 4. After the choice has been made, two r...
Find the the expected value. Thank you.
Problem 2: A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4 If the number chosen appears on the bottom of exactly one die after it has been rolled, then the player wins $1 dollar. If the number chosen appears on the bottom of both of the dice, then the...
Three six-sided fair dice are rolled. The six sides are numbered 1,2,3,4,5,6. Let A be the...
Three six-sided fair dice are rolled. The six sides are numbered 1,2,3,4,5,6. Let A be the event that the first die shows an even number, let B be the event that the second die shows an even number, and let C be the event that the third die shows an even number. Express each of the following events in terms of the named events described above: 1) the event that all three dice show even numbers 2) the event that...
(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...
Consider a die with 2 red sides, 2 green sides, and 2 blue sides. Roll the...
Consider a die with 2 red sides, 2 green sides, and 2 blue sides. Roll the die 5 times, and let X denote the number of times that the die has a red result. Flip a coin 5 times, and let Y denote the number of times that the coin shows “heads." a. Find E(X?Y). b. Find Var?(X?Y).
Problem #3: 5 fair 12-sided dice are rolled. (a) [3 marks] Find the conditional probability that...
Problem #3: 5 fair 12-sided dice are rolled. (a) [3 marks] Find the conditional probability that at least one die lands on 3 given that all 5 dice land on different numbers. 6) [2 marks] True or False: If X is the maximum of the 5 numbers from one roll, and Y is the minimum of the 5 numbers from one roll, then X and Y are independent random variables.
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z...
Let Ņ, X1. X2, . . . random variables over a probability space It is assumed that N takes nonnegative inteqer values. Let Zmax [X1, -. .XN! and W-min\X1,... ,XN Find the distribution function of Z and W, if it suppose N, X1, X2, are independent random variables and X,, have the same distribution function, F, and a) N-1 is a geometric random variable with parameter p (P(N-k), (k 1,2,.)) b) V - 1 is a Poisson random variable with...
You have a pair of 4-sided dice. The four sides of each die are numbered 1, 2. 3, and 4. Each time the pair of dice is rolled, you add the numbers from each die. Out of all the possible ways the dice can land, how many of them give you a sum of 5? Number How many ways give you a sum of 8? Number What is the probability of rolling a sum of 7 with these dice? Number
Find the the expected value. Thank you.
Problem 2: A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4 If the number chosen appears on the bottom of exactly one die after it has been rolled, then the player wins $1 dollar. If the number chosen appears on the bottom of both of the dice, then the...
(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...
Problem #3: 5 fair 12-sided dice are rolled. (a) [3 marks] Find the conditional probability that at least one die lands on 3 given that all 5 dice land on different numbers. 6) [2 marks] True or False: If X is the maximum of the 5 numbers from one roll, and Y is the minimum of the 5 numbers from one roll, then X and Y are independent random variables.
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