Question

In: Math

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.

Expert Solution

a) Possible values of X are: 0, 1, 2, 3, 4, 5

b) P(X = 0) = P(1,1) + P(2,2) + P(3,3) + P(4,4) + P(5,5) + P(6,6) = 6/36 = 1/6

P(X = 1) = P(1,2) + P(2,1) + P(2,3) + P(3,2) + P(3,4) + P(4,3) + P(4,5) + P(5,4) + P(5,6) + P(6,5) = 10/36 = 5/18

P(X = 2) = P(1,3) + P(3,1) + P(2,4) + P(4,2) + P(3,5) + P(5,3) + P(4,6) +P(6,4) = 8/36 = 2/9

P(X = 3) = P(1,4) + P(4,1) + P(2,5) + P(5,2) + P(3,6) + P(6,3) = 6/36 = 1/6

P(X = 4) = P(1,5) + P(5,1) + P(2,6) + P(6,2) = 4/36 = 1/9

P(X = 5) = P(1,6) + P(6,1) = 2/36 = 1/18

c) P(2 < X < 5) = P(X = 3) + P(X = 4) = 1/6 + 1/9 = 5/18

P(2 < X < 5) = P(X = 2) + P(2 < X < 5) = 2/9 + 5/18 = 1/2

d) Expected value, E(X) = 0 * 6/36 + 1 * 10/36 + 2 * 8/36 + 3 * 6/36 + 4 * 4/36 + 5 * 2/36 = 1.944

E(X²) = 0² * 6/36 + 1² * 10/36 + 2² * 8/36 + 3² * 6/36 + 4² * 4/36 + 5² * 2/36 = 5.833

Var(X) = E(X²) - (E(X))² = 5.833 - 1.944² = 2.0539

Standard deviation = sqrt(2.0539) = 1.43

milcah answered 1 year ago

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]

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 dice are rolled. Let the random variable X denote the number that falls uppermost on...

Two dice are rolled. Let the random variable X denote the number that falls uppermost on the first die and let Y denote the number that falls uppermost on the second die. (a) Find the probability distributions of X and Y. x 1 2 3 4 5 6 P(X = x) y 1 2 3 4 5 6 P(Y = y) (b) Find the probability distribution of X + Y. x + y 2 3 4 5 6 7 P(X...

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).

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...

You roll two fair dice, and denote the number they show by X and Y. Let...

You roll two fair dice, and denote the number they show by X and Y. Let U = min{X, Y } and V = max{X, Y }. Write down the joint probability mass function of (U, V ) and compute ρ(U, V ) i.e the correlation coefficient of U and V

A fair 4-sided die is rolled, let X denote the outcome. After that, if X =...

A fair 4-sided die is rolled, let X denote the outcome. After that, if X = x, then x fair coins are tossed, let Y denote the number of Tails observed. a) Find P( X >= 3 | Y = 0 ). b) Find E( X | Y = 2 ). “Hint”: Construct the joint probability distribution for ( X, Y ) first. Write it in the form of a rectangular array with x = 1, 2, 3, 4 and...

A pair of fair dice are rolled once. Suppose that you lose $7 if the dice...

A pair of fair dice are rolled once. Suppose that you lose $7 if the dice sum to 3 and win $12 if the dice sum to 4 or 12. How much should you win or lose if any other sum turns up in order for the game to be fair. I got a negative answer and im not sure thats correct.

let x be the random variable in rolling two dice denotes the absolute difference between die...

let x be the random variable in rolling two dice denotes the absolute difference between die faces . (i) find the probability distribution f(x) and cumulative distribution F(x) . ii) plot f(x) and F(x) iii) find the mean and variance of x and of g(x) = 3x + 7