Two fair dice are rolled and the outcomes are recorded. Let X denotes the larger of...

Two fair dice are rolled and the outcomes are recorded. Let X denotes the larger of the two numbers obtained and Y the smaller of the two numbers obtained. Determine probability mass functions for X and Y, and the cumulative distribution functions for X and for Y. Present the two cumulative distribution functions in a plot. Calculate E (2X + 2Y −8).

Expert Solution

Let X denotes the larger of the two numbers obtained . So the possible values of X are 1, 2, 3, 4, 5, and 6 with frequencies as 1, 3, 5, 7, 9, and 11 respectively.

So the probability distribution function ( f(x) ) of X and cumulative distribution function ( F(x) ) are as follows:

X	1	2	3	4	5	6	Total
P(X =x)	1/36	3/36	5/36	7/36	9/36	11/36	1
F(x)	1/36	4/36	9/36	16/36	25/36	1

Let Y denotes the smaller of the two numbers obtained . So the possible values of Y are 1, 2, 3, 4, 5, and 6 with frequencies as 11, 9, 7, 5, 3, and 1 respectively.

So the probability distribution function ( f(y) ) of Y and the cumulative distribution function ( F(y) ) of Y are as follows:

Y	1	2	3	4	5	6	Total
P(Y = y)	11/36	9/36	7/36	5/36	3/36	1/36	1
F(y)	11/36	20/36	27/36	32/36	35/36	1

Let's find E(X) and E(Y):

E(X) = 1*(1/36) + 2* (3/36) + 3*(5/36) + 4*(7/36) + 5*(9/36) + 6*(11/36) = 161/36 = 4.4722

E(Y) = 1*(11/36) + 2*(9/36) + 3*(7/36) + 4*(5/36) + 5*(3/36) + 6*(1/36) = 91/36 = 2.5278

E (2X + 2Y −8) = 2*(161/36) + 2*(91/36) - 8 = 14 - 8 = 6

milcah answered 7 months ago

Two fair dice are rolled. Let X be the product of the number of dots that...

Two fair dice are rolled. Let X be the product of the number of dots that show up. (a) Compute P(X = n) for all possible values of n. (b) Compute E(X). (c) Compute Var(X) and SD(X).

Two 6-sided dice are rolled. Let X be the larger of the two numbers showing. For...

Two 6-sided dice are rolled. Let X be the larger of the two numbers showing. For each i from 1 to 6, find the probability that X = i.

Two fair dice are rolled at once. Let x denote the difference in the number of...

Two fair dice are rolled at once. Let x denote the difference in the number of dots that appear on the top faces of the two dice. For example, if a 1 and a 5 are rolled, the difference is 5−1=4, so x=4. If two sixes are rolled, 6−6=0, so x=0. Construct the probability distribution for x. Arrange x in increasing order and write the probabilities P(x) as simplified fractions.

Suppose two fair dice are rolled. Let X denote the product of the values on the...

Suppose two fair dice are rolled. Let X denote the product of the values on the dice and Y denote minimum of the two dice. Find E[X] and E[Y] Find Var X and Var Y Let Z=XY. Find E[Z]. Find Cov(X,Y) and Corr(X,Y) Find E[X|Y=1] and E[Y|X=1]

2. Three fair dice are rolled. Let X be the sum of the 3 dice. (a)...

2. Three fair dice are rolled. Let X be the sum of the 3 dice. (a) What is the range of values that X can have? (b) Find the probabilities of the values occuring in part (a); that is, P(X = k) for each k in part (a). (Make a table.) 3. Let X denote the difference between the number of heads and the number of tails obtained when a coin is tossed n times. (a) What are the possible...

two fair dice are each rolled once. Let X denote the absolute value of the difference...

two fair dice are each rolled once. Let X denote the absolute value of the difference between the two numbers that appear. List all possible values of X Find the probability distribution of X. Find the probabilities P(2<X<5) and P(2£X<5). Find the expected value mand standard deviation of X.

Let X be the outcome of rolling a fair six-sided dice. The possible outcomes or X...

Let X be the outcome of rolling a fair six-sided dice. The possible outcomes or X are 1,2,3,4,5 and 6 and all are equally likely. What is the cumulative distribution function F(x)?

Suppose we roll two fair dice. Let D1 be the random variable that denotes the value...

Suppose we roll two fair dice. Let D1 be the random variable that denotes the value of the first dice and D2 the sum of the numbers of both dice. a) Calculate the joint mass function of D1 and D2. b) Calculate the conditional mass function of D1 given D2 = d2. c) Calculate the conditional mass function of D2 given D1 = d1. d) Are the variables D1 and D2 independent? Argue your answer.

5. Two fair, distinct dice (one red and one green) are rolled. Let A be the...

5. Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is 8. Are A,B independent events? According to the American Lung Association 7% of the population has lung disease. Of the people having lung disease 90% are smokers. Of the people not having lung disease 20% are smokers. What are the chances that a smoker has...

Two fair, distinct dice (one red and one green) are rolled. Let A be the event...

Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is even. Are A,B independent events? Please show work where applicable.

Question