Question

In: Statistics and Probability

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 + Y = x + y)
x + y 8 9 10 11 12
P(X + Y = x + y)

Solutions

Expert Solution

a)

x        P(X = x)

1            1/6

2            1/6

3            1/6

4            1/6

5            1/6

6            1/6

y        P(Y = y)

1            1/6

2            1/6

3            1/6

4            1/6

5            1/6

6            1/6

b)

x+y           P(X + Y = x+y)

2                P(1,1) = 1/6 * 1/6 = 1/36

3                P(1,2) + P(2,1) = 2 * 1/6 * 1/6 = 1/18

4                P(1,3) + P(2,2) + P(3,1) = 3 * 1/6 * 1/6 = 1/12

5                P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 * 1/6 * 1/6 = 1/9

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

7                P(1,6) + P(2,5) + P(3,4) + P(4,3) + P(5,2) + P(6,1) = 6 * 1/6 * 1/6 = 1/6

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

9                P(3,6) + P(4,5) + P(5,4) + P(6,3) = 4 * 1/6 * 1/6 = 1/9

10              P(4,6) + P(5,5) + P(6,4) = 3 * 1/6 * 1/6 = 1/12

11              P(5,6) + P(6,5) = 2 * 1/6 * 1/6 = 1/18

12              P(6,6) = 1/6 * 1/6 = 1/36


Related Solutions

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.
Two dice are rolled. Let X be the random variable representing the result of the first...
Two dice are rolled. Let X be the random variable representing the result of the first die, and Y be the random variable representing the largest value rolled on either die. Describe the joint probability density function for X and Y .
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 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.
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).
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
2 dice are rolled. Let X be the number on the first die, Y - on...
2 dice are rolled. Let X be the number on the first die, Y - on the second, and Z=X - Y. Find the expectation and standard deviation of Z.
An unbiased coin is tossed four times. Let the random variable X denote the greatest number...
An unbiased coin is tossed four times. Let the random variable X denote the greatest number of successive heads occurring in the four tosses (e.g. if HTHH occurs, then X = 2, but if TTHT occurs, then X = 1). Derive E(X) and Var(X). (ii) The random variable Y is the number of heads occurring in the four tosses. Find Cov(X,Y).
a die is rolled 6 times let X denote the number of 2's that appear on...
a die is rolled 6 times let X denote the number of 2's that appear on the die. 1. show that X is binomial. 2. what is the porbaility of getting at least one 2. 3. find the mean and the standard deviaion of X
Consider rolling two dice and let (X, Y) be the random variable pair defined such that...
Consider rolling two dice and let (X, Y) be the random variable pair defined such that X is the sum of the rolls and Y is the maximum of the rolls. Find the following: (1) E[X/Y] (2) P(X > Y ) (3) P(X = 7) (4) P(Y ≤ 4) (5) P(X = 7, Y = 4)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT