A fair tetrahedron (four-sided die) is rolled twice. Let X be the random variable denoting the...
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 PMF.
(c) Find the PMF of Y and present it graphically. Describe its shape. How does it differ from that of X?
(d) Compute the means, variances, and standard deviations of X and Y .
(e) Obtain the joint probability mass function of (X, Y ). Present in tabular form and if possible in graphical form.
(f) Obtain the covariance and correlation between (X, Y ).
(g) Are X and Y independent or dependent?
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A fair tetrahedron (four-sided die) is rolled twice. Let X be
the random variable denoting the...
A tetrahedron (four-sided die) is rolled twice and the sum X of k is the results...
A tetrahedron (four-sided die) is rolled twice and the sum X of k is the results of the two rolls is recorded. We know that the chance that X proportional to k. (a) What is the probability model for X, i.e., uhat values can X take, and what are the corresponding probabilities? (b) Compute the chance that the sum of the two rolls will erceed but ill not be more than b (c) Compute the erpected value of x.
A die is rolled. Assume that a random variable X represents the outcomes of this experiment
Example #2: A die is rolled. Assume that a random variable X represents the outcomes of this experiment. Construct a probability distribution table and represent this probability distribution graphically. (Use the x-axis for values of X and the y-axis for P(X)). Example #3: A coin is tossed 3 times. Suppose that the random variable X is defined as the number of heads. Construct a probability distribution of X and represent this probability distribution graphically. (Use the x-axis for values of X and the...
Problem 2. A tetrahedron four-sided die) is rolled turice and the sum X of the results...
Problem 2. A tetrahedron four-sided die) is rolled turice and the sum X of the results of the two rolls is recorded. We know that the chance that X-k is proportional to k. (a) What is the probability model for X, i.e., what values can X take, and what are the corresponding probabilities? (b) Compute the chance that the sum of the two rolls wieceed but will not be more than 6. (c) Compute the expected value of 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...
Exercise 5.11. Suppose a fair 1-sided die is rolled, and the random variable X (s) outputs...
Exercise 5.11. Suppose a fair 1-sided die is rolled, and the random variable X (s) outputs - 1 if the roll is 2, and 1 if the roll is 1,3, or 1. Calculate Mx(2).
Consider three six-sided dice, and let random variable Y = the value of the face for...
Consider three six-sided dice, and let random variable Y = the value of the face for each. The probability mass of function of Y is given by the following table: y 1 2 3 4 5 6 otherwise P(Y=y) 0.35 0.30 0.25 0.05 0.03 0.02 0 Roll the three dice and let random variable X = sum of the three faces. Repeat this experiment 50000 times. Find the simulated probability mass function (pmf) of random variable X. Find the simulated...
Consider a fair six-sided die. (a) What is its probability mass function? Graph it. It represents...
Consider a fair six-sided die. (a) What is its probability mass function? Graph it. It represents the population distribution of outcomes of rolls of a six-sided die (b) How would you describe the population distribution? (c) What is the sampling distribution of x for a six-sided fair die, when its rolled 100 times? Describe it with as much specificity as possible. NOTE: Roll of a die is a discrete variable. Why is it ok to use the Normal distribution to...
Roll two fair four-sided dice. Let X and Y be the die scores from the 1st...
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?
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...
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 X,Y. (b) The pairs inside the matrix corresponding to a fixed value of X - Y form a straight line of entries inside the matrix. Draw those lines and use them to construct the probability mass function of the random variable X-Y- make...
(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.
A tetrahedron (four-sided die) is rolled twice and the sum X of k is the results of the two rolls is recorded. We know that the chance that X proportional to k. (a) What is the probability model for X, i.e., uhat values can X take, and what are the corresponding probabilities? (b) Compute the chance that the sum of the two rolls will erceed but ill not be more than b (c) Compute the erpected value of x.
Problem 2. A tetrahedron four-sided die) is rolled turice and the sum X of the results of the two rolls is recorded. We know that the chance that X-k is proportional to k. (a) What is the probability model for X, i.e., what values can X take, and what are the corresponding probabilities? (b) Compute the chance that the sum of the two rolls wieceed but will not be more than 6. (c) Compute the expected value of X
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...
Exercise 5.11. Suppose a fair 1-sided die is rolled, and the random variable X (s) outputs - 1 if the roll is 2, and 1 if the roll is 1,3, or 1. Calculate Mx(2).
Consider a fair six-sided die. (a) What is its probability mass function? Graph it. It represents the population distribution of outcomes of rolls of a six-sided die (b) How would you describe the population distribution? (c) What is the sampling distribution of x for a six-sided fair die, when its rolled 100 times? Describe it with as much specificity as possible. NOTE: Roll of a die is a discrete variable. Why is it ok to use the Normal distribution to...
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 X,Y. (b) The pairs inside the matrix corresponding to a fixed value of X - Y form a straight line of entries inside the matrix. Draw those lines and use them to construct the probability mass function of the random variable X-Y- make...
(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.
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