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Let Yn be the total number of spots obtained in n independent rolls of a 6-face...
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.
Problem 1. Suppose that a fair six-sided die is rolled n times. Let N be the number of 1's rolled...
I know Pk~1/k^5/2 just need the
work
Problem 1. Suppose that a fair six-sided die is rolled n times. Let N be the number of 1's rolled, N2 be the number of 2's rolled, etc, so that NN2+Ns-n Since the dice rolls are independent then the random vector < N,, ,Ne > has a multinomial distribution, which you could look up in any probability textbook or on the web. If n 6k is a multiple of 6, let Pa be...
7. In n rolls of a fair die, let X be the number of times 1 is rolled, and Y the number of times ...
7. In n rolls of a fair die, let X be the number of times 1 is rolled, and Y the number of times 2 is rolled. Find the conditional distribution of X given Y-m
7. In n rolls of a fair die, let X be the number of times 1 is rolled, and Y the number of times 2 is rolled. Find the conditional distribution of X given Y-m
We roll a fair die repeatedly. Let N be the number of rolls needed to see...
We roll a fair die repeatedly. Let N be the number of rolls needed to see the first six, and let Y be the number of fives in the first N -1 rolls. In class, we saw that E[Y I N]- (N - 1)/5. Using this, find EiY]. Also, find Cov(Y, N). Hint: N -1 is a geometric random variable. (Why?)
Suppose that Adam rolls a fair six-sided die and a fair four-sided die simultaneously. Let A...
Suppose that Adam rolls a fair six-sided die and a fair four-sided die simultaneously. Let A be the event that the six-sided die is an even number and B be the event that the four-sided die is an odd number. Using the sample space of possible outcomes below, answer each of the following questions.What is P(A), the probability that the six-sided die is an even number?What is P(B), the probability that the four-sided die is an odd number?What is P(A...
(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...
(3.) A fair six-sided die is rolled repeatedly. Let R denote the random variable representing the...
(3.) A fair six-sided die is rolled repeatedly. Let R denote the random variable representing the outcome of any particular roll. The following random variables are all discrete-time Markov chains. Specify the transition probabilities for each (as a check, make sure the row sums equal 1) (a) Xn, which represents the largest number obtained by the nth roll. (b) Yn, which represents the number of sixes obtained in n rolls.
i. Consider a weighted 6-sided biased die that is twice as likely to produce any even...
i. Consider a weighted 6-sided biased die that is twice as likely to produce any even outcome as any odd outcome. What is the expected value of 1 roll of this die? What is the expected value of the sum of 9 rolls of this die? ii. Let X denote the value of the sum of 10 rolls of an unweighted 6-sided die. What is Pr(X = 0 mod 6)? (Hint: it is sufficient to consider just the last roll)...
independent. Let Sa be the number of A fair coin is tossed n times. Assume the...
independent. Let Sa be the number of A fair coin is tossed n times. Assume the n trials are lower bound on the probability that heads obtained. Using Chebyshev's inequality, find a differs from 0.5 by less than 0.1 when n = 10,000. How many trials are needed to ensure that this lower bound exceeds 0.999 ?
independent. Let Sa be the number of A fair coin is tossed n times. Assume the n trials are lower bound on the...
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that...
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that there is no three consecutive successes in n trails. (1). Show that limn+cQn = 0 (2). Show that Qn = (1 - pQn-1 + p(1 - pQn-2 + p (1 - p)Qn-3 for n 3 (Hint: condition on the first failure). Problem 5. Suppose we do n independent trails that each has a probability P E (0,1) to result in success. Let Pn be...
I know Pk~1/k^5/2 just need the
work
Problem 1. Suppose that a fair six-sided die is rolled n times. Let N be the number of 1's rolled, N2 be the number of 2's rolled, etc, so that NN2+Ns-n Since the dice rolls are independent then the random vector < N,, ,Ne > has a multinomial distribution, which you could look up in any probability textbook or on the web. If n 6k is a multiple of 6, let Pa be...
7. In n rolls of a fair die, let X be the number of times 1 is rolled, and Y the number of times 2 is rolled. Find the conditional distribution of X given Y-m
7. In n rolls of a fair die, let X be the number of times 1 is rolled, and Y the number of times 2 is rolled. Find the conditional distribution of X given Y-m
We roll a fair die repeatedly. Let N be the number of rolls needed to see the first six, and let Y be the number of fives in the first N -1 rolls. In class, we saw that E[Y I N]- (N - 1)/5. Using this, find EiY]. Also, find Cov(Y, N). Hint: N -1 is a geometric random variable. (Why?)
(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...
(3.) A fair six-sided die is rolled repeatedly. Let R denote the random variable representing the outcome of any particular roll. The following random variables are all discrete-time Markov chains. Specify the transition probabilities for each (as a check, make sure the row sums equal 1) (a) Xn, which represents the largest number obtained by the nth roll. (b) Yn, which represents the number of sixes obtained in n rolls.
independent. Let Sa be the number of A fair coin is tossed n times. Assume the n trials are lower bound on the probability that heads obtained. Using Chebyshev's inequality, find a differs from 0.5 by less than 0.1 when n = 10,000. How many trials are needed to ensure that this lower bound exceeds 0.999 ?
independent. Let Sa be the number of A fair coin is tossed n times. Assume the n trials are lower bound on the...
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that there is no three consecutive successes in n trails. (1). Show that limn+cQn = 0 (2). Show that Qn = (1 - pQn-1 + p(1 - pQn-2 + p (1 - p)Qn-3 for n 3 (Hint: condition on the first failure). Problem 5. Suppose we do n independent trails that each has a probability P E (0,1) to result in success. Let Pn be...
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