Problem 3 Roll a die until we get a 6. Let X be the total number of rolls and Y the number of l's we get. (a) Find Etx Y k (b) Find EY Problem 3 Roll a die until we get a 6. Let X be 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...
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?)
Problem 1. Suppose that you roll an 8-sided die until you get an 8. Let G denote the number of rolls that this takes. (a) Write down the probability mass function of G (b) Write a closed-form expression for P(G 2 6) (i.e. do not just write it as an infinite sum) (c) Do calculations to show that P(G2 10 | G> 6) P(G2 4). Problem 1. Suppose that you roll an 8-sided die until you get an 8. Let...
If a die is rolled six times, let X be then number the die obtained on the first roll and Y be the sum of the numbers obtained from all the rolls. Find the expected value and variance of x and y.
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
help with number 4 4. Roll a die and flip a coin. Let Y be the value of the die. Let Z = 1 if the coin shows a head, and Z = 0 otherwise. Let X = Y + Z. Find the variance of X. 5. (a) If X is a Poisson random variable with = 3, find E(5*).
We roll a fair 8-sided die five times. (A fair 8-sided die is equally likely to be 1, 2, 3, 4, 5, 6, 7, or 8.) (a) What is the probability that at least one of the rolls is a 3? (b) Let X be the number of different values rolled. For example, if the five rolls are 2, 3, 8, 8, 7, then X = 4 (since four different values were rolled: 2,3,7,8). Find E[X].
problem 4 you repeatedly roll an ordinary eight-sided die five times. Let X equal the number of times you roll the die. Let Y equal the value of the first roll What is E[x] and E[Y]
Problem 6. A fair die is rolled four times. (a) Let Y denote the number of distinct rolls. Find the probability mass function of Y. (b) Let Z denote the minimal result fo the 4 throws. Find the probability mass function of Z
We flip a coin. If it is heads we roll a four sided die with sides numbered from 1 to 4. If it is tails, we roll a six sided die with sides numbered from 1 to 6. We let X be the number rolled. (a) What is the expectation of X? (b) What is the variance of X? (c) What is the standard deviation of X? We draw cards one by one and with replacement from a standard deck...