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...
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.
(20pts) Problem 3. A pair of fair dice are cast, and the number of rolled dots, on each die, is recorded. Let X denote the difference of the two numbers (10pts)a. Find the probability mass function of X. b. Find the expected value E(X).
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...
4. Two fair dice are rolled. Compute the probability that the number on the first is smaller than the number on the second
Please answer the question clearly 6. Suppose that we roll a pair of balanced dice. Let X be the number of dice that show 1 and Y be the number of dice that show either 4, 5, or 6. (a) Draw a diagram like Figure 3.1 on page 62 showing the values of the pair (X, Y) associated with each of the 36 equally likely points of the sample space. (b) Construct a table showing the values of the joint...
3. A fair die is rolled twice. Let the first outcome be X and the second outcome be Y (a) (5 points) Calculate Cov(XY, X -Y) and simplify. (hint: What is Cov(aX+bY) in terms of Cov(X, Y)?) (b) (5 points) Are X Y and X -Y independent? Explain. (c) (5 points) Calculate the moment generating function Mx+y(t) of X+Y (the answer should be a function of t and can contain unsimplified sums)
Two four-sided dice, one red and one white, will be rolled. List the possible values for the following random variable. Let Y = difference between the number on the red die and the number on the white die (red-white). a. Draw the probability histogram for this random variable. b. What is the most likely value of Y? c. What is the probability that the difference on the dice is negative? Use proper notation throughout. Write your answer as a decimal.
A fair tetrahedron (four-sided die) is rolled twice. Let X be the random variable denoting the total number of dots in the outcomes, and Y be the random variable denoting the maximum in the two outcomes. Thus if the outcome is a (2, 3) then X = 5 while Y = 3. (a) What are the ranges of X and Y ? (b) Find the probability mass function (PMF) of X and present it graphically. Describe the shape of this...
For the two six-sided dice case: Write out the six-by-six matrix showing all possible (36) combinations of outcomes. Draw a histogram of the probability of outcomes for the dice totals. Explain the shape of the histogram. Draw a Venn diagram for the 36 dice roll combinations. Define a set "A" as all the combinations that total seven; define set "B" as all the combinations that have one die roll (either die 1 or 2) equal to 2. Indicate the sets...
Problem 5 Two fair dice are tossed, and ((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 . (A) Derive the distribution of U. List all possible values and evaluate their probabilities. (B) Derive the distribution of W. List all possible values and evaluate their probabilities. (C) Determine the conditional probability P (6 SU <71W <1]