Question

In: Statistics and Probability

Let Y be the sum of two fair six-sided die. (a) Find the PMF of Y....

Let Y be the sum of two fair six-sided die.

(a) Find the PMF of Y.

(b) What is the expected value of Y ?

(c ) What is the standard deviation of Y ?

(d) Interpret the standard deviation you found in the last part in context of the experiment.

(e) Find the CDF of Y.

(f) Use the CDF of Y to find the probability that the sum of the dice will be strictly between six and ten. Answers calculated using the PDF will receive no credit.

Expert Solution

Let Y be the sum of two fair six-sided die.

a. The PMF of Y is given by:

Y=y P(Y=y)

2 1/36

3 2/36

4 3/36

5 4/36

6 5/36

7 6/36

8 5/36

9 4/36

10 3/36

11 2/36

12 1/36

b) The expected value of Y is given by:

c) The standard deviation of Y is given by:

Where,

Therefore

d) The standard deviation of Y indicate that the spread of the variable is 2.39 from the mean.

That is the sum of the faces of two fair dice varies by 2.39 from the mean value of 7.

e) The CDF of Y is given by

Y=y P(Y=y) P(Y<=y)

2 1/36 1/36

3 2/36 1/36+2/36=3/36

4 3/36 3/36+3/36=6/36

5 4/36 6/36+/36=10/36

6 5/36 10/36+5/36=15/36

7 6/36 15/36+6/36=21/36

8 5/36 21/36+5/36=26/36

9 4/36 26/36+4/36=30/36

10 3/36 30/36+3/36=33/36

11 2/36 33/36+2/36=35/36

12 1/36 35/36+1/36=36/36=1

e) the probability that the sum of the dice will be strictly between six and ten is given by

orchestra answered 1 year ago

A regular six-sided die and a regular eight-sided die are rolled to find the sum. Determine...

A regular six-sided die and a regular eight-sided die are rolled to find the sum. Determine the probability distribution for the sum of the two dice. Create a frequency histogram for the probability distribution and determine the expected sum of the two dice.

Suppose you are rolling a fair four-sided die and a fair six-sided die and you are...

Suppose you are rolling a fair four-sided die and a fair six-sided die and you are counting the number of ones that come up. a) Distinguish between the outcomes and events. b) What is the probability that both die roll ones? c) What is the probability that exactly one die rolls a one? d) What is the probability that neither die rolls a one? e) What is the expected number of ones? f) If you did this 1000 times, approximately...

Find the probability that in 200 tosses of a fair six sided die, a five will...

Find the probability that in 200 tosses of a fair six sided die, a five will be obtained at least 40 times

Let ? be the number that shows up when you roll a fair, six-sided die, and,...

Let ? be the number that shows up when you roll a fair, six-sided die, and, let ? = ?^2 − 5? + 6. a. Find both formats for the distribution of ?. (Hint: tep forms of probability distributions are CDF and pmf/pdf.) b. Find F(2.35)

A regular six-faced fair die will be rolled 144 times. Let X be the sum of...

A regular six-faced fair die will be rolled 144 times. Let X be the sum of the 144 numbers obtained. Find the approximate probability that X is between 463 and 545.

A fair six-sided die is rolled repeatedly until the third time a 6 is rolled. Let...

A fair six-sided die is rolled repeatedly until the third time a 6 is rolled. Let X denote the number of rolls required until the third 6 is rolled. Find the probability that fewer than 5 rolls will be required to roll a 6 three times.

Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls...

Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls it takes to obtain the first 5, Y denote the number of rolls until the first 2, and Z denote the number of rolls until the first 4. Numerical answers are needed only for parts (a) and (b). Expressions are sufficient for parts (c), (d), and (e). a) E[X|Y = 1 or Z = 1] b) E[X|Y = 1 and Z = 2] c)...

Suppose a red six-sided die, a blue six-sided die, and a yellow six-sided die are rolled....

Suppose a red six-sided die, a blue six-sided die, and a yellow six-sided die are rolled. Let - X1 be the random variable which is 1 if the numbers rolled on the blue die and the yellow die are the same and 0 otherwise; - X2 be the random variable which is 1 if the numbers rolled on the red die and the yellow die are the same and 0 otherwise; - X3 be the random variable which is 1...

17#13 Suppose we roll a fair six-sided die and sum the values obtained on each roll,...

17#13 Suppose we roll a fair six-sided die and sum the values obtained on each roll, stopping once our sum exceeds 289. Approximate the probability that at least 76 rolls are needed to get this sum.

Two rolls of a fair die. Let x and y be the results of the two...

Two rolls of a fair die. Let x and y be the results of the two rolls, and let z = x + y. (a) Find P[x = 4, y = 3], P[x > 3, y = 5]. (b) Find P[z = 7], P[z = 5], P[z = 3]. (c) Find the probability that at least one 6 appears given that z = 8. (d) Find the probability that x = 6 given that z > 8. (e) Find the...