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 table.)
(b) Find the marginal PMF of X.
(c) Compute the mean and variance of X.
(d) Find the conditional PMF of Y given X = 1.
(e) Compute the conditional mean and variance of Y given X = 1.
a) The joint PDF for X, Y is obtained here as:
X = 0 | X = 1 | X = 2 | |
Y = 0 | 0 | 0 | (1/3)4 = 1/81 |
Y = 1 | 0 | (2c1)*(2/3)*(1/3)*(1/3) = 4/27 | (1/3)3(2/3)*2 = 4/81 |
Y = 2 | (2/3)2 = 4/9 | (2c1)*(2/3)*(1/3)*(2/3) = 8/27 | (1/3)2*(2/3)2 = 4/81 |
b) The marginal PDF from above joint PDF now could be obtained as:
P(X = 0) = 4/9
P(X = 1) = 12/27 = 4/9
P(X = 2) = 1/9
P(Y = 0) = 1/81
P(Y = 1) = 16/81
P(Y = 2) = 64/81
c) The mean of X is computed as:
E(X) = 1*4/9 + 2/9 = 6/9 = 2/3
E(X2) = 1*4/9 + 22/9 =
8/9
Var(X) = E(X2) - [E(X)]2 = (8/9) - (2/3)2 = 4/9
d) P(Y = 0 | X = 1) = 0
P(Y = 1 | X = 1) = 4/12 = 1/3
P(Y = 2 | X = 2) = 2/3
e) E(Y | X = 1) = 1*(1/3) + 2*(2/3) = 5/3
E(Y2 | X = 1) = 1*(1/3) + 22*(2/3) = 3
Var(Y | X = 1) = 3 - (5/3)2 = 2/9
Two identical fair 6-sided dice are rolled simultaneously. Each die that shows a number less than...
In this experiment, both a fair four-sided die and a fair six-sided die are rolled (these dice both have the numbers most people would expect on them). Let Z be a random variable that represents the absolute value of their difference. For instance, if a 4 and a 1 are rolled, the corresponding value of Z is 3. (a) What is the pmf of Z? (b) Draw a graph of the cdf of Z
Roll two fair four-sided dice. Let X and Y be the die scores from the 1st die and the 2nd die, respectively, and define a random variable Z = X − Y (a) Find the pmf of Z. (b) Draw the histogram of the pmf of Z. (c) Find P{Z < 0}. (d) Are the events {Z < 0} and {Z is odd} independent? Why?
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.
3. Two fair, four-sided dice are rolled. Let X1, X2 be the outcomes of the first and second die, respectively. (a) Find the conditional distribution of X2 given that Xi + X2 = 4. (b) Find the conditional distribution of X2 given that Xi + X2-5.
1. Suppose 7 dice are rolled. The dice are 6-sided and fair. a). Find the probability that more than 5 dice show 2 or less (you may leave your answer in unsimplified form). I found this answer to be 5/729= 0.006859 b). Suppose we roll 7 dice and count the number showing 2 or less. We repeat this experiment over and over, each time counting the number showing 2 or less. What should we expect to compute as an average...
Find the conditional probability, in a single roll of two fair 6-sided dice, that neither die is a three, given that the sum is greater than 6 7 The probability is 12 (Type an integer or a simplified fraction)
If we roll a red 6-sided die and a green 6-sided die (both are fair dice with the numbers 1-6 equally likely to be rolled), what is the probability that we get (i) A 5 on the green die AND a 3 on the red die? (ii) A 5 on the green die OR a 3 on the red die? (iii) A 5 on the green die GIVEN we rolled a 3 on the red die?
A) Suppose I roll two fair six-sided dice. What is the probability that I rolled a total of 5? B) Suppose I roll two fair six-sided die and I announce that the sum of the two die is 6 or less. What is the probability that I rolled a total of 5?
Problem 4. Two four-sided dice are rolled simultaneously. (a) Let X be the sum of the two rolls. Calculate the PMF and the expected value of . (b) Someone proposes to give you in dollars five times the amount of the sum X that you roll, if you pay A dollars in advance. What should be the amount A in order for you to expect to break even? (c) Repeat parts (a) and (b) for the case where X is...
1. A blue fair 6-sided dice and a red fair 6-sided dice are rolled at the same time. a) What is the probability of the sum of the dice equals 7, given 1 2 3 4 5 6 at least one of the dice shows a 3? 1 (1.1) (1.2) (1.3) (1.4) (1.5) (1.6) 2 (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) 5 (5.1) (5.2) (5.3) (5.4) (5.5) (5.6)...